Identifiability Guarantees for Causal Disentanglement from Purely Observational Data

Thank you for your detailed review! We appreciate that you found our proofs sound and our newly defined notion of identifiability to be interesting. We would like to address your comments below:

“What do you exactly mean by causal disentanglement?”

Following this review (https://arxiv.org/abs/2206.15475), we define causal disentanglement to be the process of learning about latent variables $Z$ from the set of observational variables $X$, such that $X = g(Z)$ for some mapping $g$, and $Z$ factorizes as $p(Z) = \prod_{i=1}^n p(Z_i | Z_{pa(i)}, \epsilon_i)$, where $\epsilon_i$ is an exogenous noise term associated with $Z_i$. We additionally note that this problem has also been called “causal representation learning” in the literature. We will refine Section 2 to make this more clear.

“you cite [32] … there is no level corresponding to H -- if this is the novely, then please be more precise.”

“a more detailed discussion on delineating the contribution ...”

Thank you for your suggestion. We would like to clarify that prior score-based methods (including [32]) consider the problem of causal structure learning when all causal variables are fully observed, while we consider the more difficult problem of causal disentanglement, where the causal variables are inherently latent and can only be viewed through an unknown mixing function, which we denote by $H$. Using the notation defined in our paper, [32] would assume direct access to the latent variables, $Z$, for which we derive methods to learn representations of. We will add a more thorough discussion in Section 1.1 indicating this distinction between our contributions.

“assuming an ANM is a structural assumption…”

We wanted to distinguish structural assumptions from functional assumptions as restrictions to just the mapping from the latent to the observed space as defined by the mixing function. For clarity, previous works in purely observational causal disentanglement rely on structural assumptions on this mapping, the most common one being the pure child assumption ([11,14,19,52,53] in the manuscript), assuming that each latent variable has multiple observational variables for which they are the only parent. We in contrast allow any observational variable to be determined by any subset of latent variables. We briefly discussed this in lines 75-81, but we will revise this discussion to make it more clear.

“optimizing the Jacobian is computationally very expensive; I haven't found ...”

Thank you for this comment. We acknowledge that optimizing the Jacobian in Eq (1) can be computationally laborious. We briefly discussed this difficulty in lines 217-220 in Section 4.1, which is followed by our proposed solution. We also discussed the difficulty of estimating the Jacobian in lines 275-281 in Section 5.1. We will add pointers to these places in the introduction.

“..encourage the authors to address the (limitations of) identifiability up to upstream layers …”

“Can you use insights … to improve your identifiability…”

Thank you for your suggestion. The primary limitation of identifiability up to upstream layers is that variables which are more downstream in the underlying causal graph have weaker identifiability guarantees. Conversely, variables that are more upstream in the causal graph, or more influential in other words, have stronger identifiability guarantees. As shown in Section 3.3, these identifiability results can not be improved without further assumptions. It is currently unclear what minimal additional assumptions are required to achieve stronger identifiability guarantees. We note this as an important question for future work, and we will explicitly add this discussion to Section 6.

“What exactly do you mean by asymmetries?”

Asymmetries refers to the asymmetric relationships between causes and effects present in this model. More specifically, we will have Gaussian residuals if we regress the effect over causes, but not the other way around. These relationships are useful in establishing causal directions, which we utilize to learn the causal layers of the underlying graph.

“Why do you consider Gaussian exogenous variables when that causes …”

The Gaussian noise assumption is common for nonlinear additive noise models given their nice theoretical properties that allow for the whole causal graph to be learned when all causal variables are observed (c.f., [33] referred in the manuscript). Our theoretical proofs similarly rely on this Gaussianity assumption, which is why we consider them even though they cause the specified impossibility result. Considering whether stronger identifiability results could be achieved under non-Gaussianity assumptions is an interesting direction for future work.

“In Fig. 4, what is the MAC for SSM and STEIN? …”

The text “SSM” and “STEIN” in Figure 4 are intended to be labels of the black and red vertical lines indicating these methods’ average signal-to-error ratio when used to estimate the pointwise Jacobians of the score, which is 6 and 2 dB respectively. We will make these labels more clear with a legend in our revision. The MAC of the exogenous noise estimates using these estimation methods for various sample sizes is depicted in Table 2.

“… relate to block… content and style latent variables”

Since identifiability up to layers means that each causal variable can be identified up to transformation of itself and all variables in or above its layer, this means that we can achieve block-identifiability of each variable and all variables in or above its layer. In particular, this means the root variables can be identified up to their own block. For content and style variables, as content variables are in the layers above style variables, they can be identified up to their own block.

Thank you for the minor points as well. We will revise accordingly.

Thank you for the discussion and for updating the scores!

Thanks for the suggestion! We will revise the manuscript to clarify the concepts regarding (1) causal representation learning and causal disentanglement, and (2) structural restrictions and graphical restrictions. Our usage and initial thoughts aim to align with recent prior works in this area (e.g., the usage of causal disentanglement in [1], and structural restrictions in [2]). Nonetheless, we will clarify these concepts to improve readability.

In addition, we would like to comment on why we believe that while ANM poses certain assumptions, it can still be a useful model in many scenarios:

The Additive Noise Model Assumption

When compared to alternative assumptions (e.g., independent components, which are a special case of ANM) in identifiable representation learning, we believe that ANM is relatively less restrictive, as it allows dependencies between the latent components. These dependencies are important, in our opinion, as the latent factors should be related to each other. Additionally, in the causal structure learning setting where all causal variables are observed, ANMs are frequently used because (1) their theoretical properties are well understood, (2) efficient methods exist for learning them, and (3) they can fit non-parametric relationships, making them flexible for modeling many real-world causal systems, such as gene regulatory networks [3]. Therefore we believe that it would be helpful to devise understandings and methods to extend these models to the causal representation learning setting, where we can handle perceptual and non-tabular observations.

References:

[1] Causal Machine Learning: A Survey and Open Problems
[2] Linear causal disentanglement via interventions
[3] Estimation of genetic networks and functional structures between genes by using Bayesian networks and nonparametric regression

Thank you for appreciating our problem setting and for recognizing the novelty of our defined notion of identifiability. We would like to address your concerns and questions below:

“L184/L284 claims there are no structural results on the mixing function, however assumption 1 assumes that the mixing function is linear … ”

To clarify, we consider the assumption of linear mixing between the latent and observed variables to be a functional assumption rather than a structural restriction, which refers to hard limits on the sets of latent variables that can determine each observational variable. In our setting, we allow any subset of latent variables to determine any observational variable. This is in contrast to previous works in purely observational causal disentanglement, which rely on the pure child assumption ([11,14,19,52,53] in the manuscript), assuming that each latent variable has multiple observational variables for which they are the only parent.

We note that we chose to assume a linear mixing as it is essential to the proofs of our theoretical guarantees. However, our results additionally hold when the true mixing function can be reduced to linear mixing, such as in the case of polynomials being reduced to linear mappings (c.f., [2,57] in the manuscript).

“L81-82 says "our work is the first to establish identifiability guarantees in the purely observational setting without imposing any structural assumptions over the mixing function". However, the prior work [2] …”

Thank you for raising this concern. The setting considered in [2] is very different from the problem setting in our paper. While [2] considers the purely observational setting, their results are dependent on the assumption that the latent variables are distributed according to a Gaussian mixture model, which does not specifically model the causal graph, and does not encapsulate the nonlinear additive Gaussian noise model that we consider. Therefore, the results in [2] are not transferable to our setting, and our original claim in L81-82 holds since we consider the setting of causal disentanglement. We will refine the claim in lines 81-82 to clarify this.

“This work seems to be an extension of [3] where an additional linearity is added… describe the main differences in the assumptions and/or additional difficulties in the proof?”

You are correct that our work is an extension of [3]. However, the important distinction between our works is that [3] considers the setting where the causal variables are fully observed, while we consider the more difficult setting where the causal variables are latent and can only be viewed through an unknown mixing function, which is where the additional difficulties lie.

The proofs in [3] utilize variance properties on the diagonal elements of the Jacobian over the score of the causal variables to derive a topological ordering. While we utilize this result, it is not sufficient given we don't have access to the specified causal variables. In Lemma 1, we prove that we can only estimate this desired Jacobian up to an unknown quadratic parameterized by the matrix $\beta$, where $J_{\hat{Z}} (\hat{z}) = \beta^{T} J_Z(z) \beta$. We therefore must derive additional information about the whole Jacobian, not just the diagonal as in [3]. In Lemma 2, we derive properties of the estimator for the unknown matrix $\beta$ such that $J_{\hat{Z}} (\hat{z})$ has zero variance terms in its diagonal. We then take it a step further in Lemma 3 and Theorem 1, to demonstrate how these properties on $\beta$ allow us to derive layer-wise representations, by maximizing the number of zero-variance terms in the diagonal of the Jacobian. This gives rise to a simple principle in Eq (1) to achieve identifiability, which serves as foundation of our algorithm.

“Experiments are on synthetic data .. an application to real-life would be needed to see if these ideas extend to practice.”

We thank the reviewer for this comment. Similar to [46,18,9] in our manuscript, we view this paper as having a primarily theoretical contribution, in which our experiments on synthetic data provide a convincing proof-of-concept for our main results.

We acknowledge the importance of real-world experiments and do believe there are many real-world scenarios, such as topic modeling in natural language for example, that could be interesting settings to evaluate our methods. In particular, the method could be used to identify hierarchical topics at different layers of the underlying causal structure. Such experiments would however necessitate an extensive amount of additional thought in experimental set up, which we believe is out of the scope of our current work, which mainly focuses on theoretical guarantees.

“Identifiability up to layers seems a bit hard to grasp… what does it mean for the practitioner?”

For causal systems which have an inherent hierarchical structure, upstream layer representations will capture all of the information used in determining each particular level of variables. This can be best explained with an example. Consider we wish to disentangle information about a latent genealogical tree, which we don’t have the ability to intervene on as it is no longer active. Intuitively, each layer representation would contain all of the prior ancestral information used to determine a given generation's traits, where the top layer would only contain information about the original ancestors and the bottom layer representation would contain all hereditary information. The additional layer-wise noise representations would capture each generation’s level of exogenous noise, providing an understanding of how traits could have spawned over time. We will add this particular example to the introduction to build intuition for how our work can be useful beyond just the theoretical guarantees.

“Typo in L26: extend -> extent”

Thank you for identifying this. We will revise accordingly.

Thank you for your response. I understand the author’s clarifications on the differences between the works that I have cited, but my concern is more with the framing of the contributions and it still seems like overselling of the work even if there are clear technical differences from prior works. While there is some intuition behind identifiability upto layers that the authors give, I still find it hard to understand why it would be useful either theoretically or in practice. Thanks for the other technical clarifications as well, I have increased my score.

Thank you for your encouraging review! We appreciate you recognizing both the importance of our work and the merit of our theoretical findings and algorithmic approach. We would like to address your additional comments and questions below:

“…this paper assumed non-linear function with Gaussian additive noise and linear mixing, but these assumptions cannot be tested from observations, either. In other words, these assumptions are, in my view alternative assumptions on the same problem, but do not relax the previous assumptions.”

We thank the reviewer for this comment, and we take this opportunity to give more intuition on our model assumptions and clarify the difference between them and assumptions on interventions. Although it may seem like a strong assumption to consider a non-linear additive Gaussian noise model as the representation of the underlying causal network, we note that this model type is frequently assumed in causal inference given their theoretical properties are well understood and there exist many methods to learn their structure in the fully observable setting [1-4]. Additionally, these models are known to be more flexible than linear additive noise models given their ability to fit non-parametric relationships, making them a commonly assumed model for many real-world causal systems such as gene regulatory networks [5] without needed verification. For the mixing function, we chose to assume a linear mixing as it is essential to the proofs of our theoretical guarantees. However, our results also hold when the true mixing function can be reduced to linear mixing using existing techniques, such as in the case of polynomials being reduced to linear mappings (c.f. [6,8]). It would be interesting in future work to understand to what other settings our results can be extended.

We recognize that there are different levels of assumptions made on the latent model and the mixing function, but we do not consider access to interventions as alternative assumptions on the same problem. While our assumptions are restricting the model class of the data-generating process, the assumption of interventions assume existence of additional data / environments, which is intrinsically different. This is also why we separate and highlight the assumption on “data” from “latent model” and “structural mixing” in Table 1 of our paper. However, we want to clarify that we do not consider our work as a strict relaxation of prior works, as the works which assume interventions usually compare multiple environments and might arrive at stronger identifiability results by imposing assumptions on the number / types of interventions.

“Besides, in this setting, the dimension of latent factors is assumed to be known, which may be a limitation.”

Thank you for this comment. We would like to clarify that we do not assume the dimension of the latent vector is known. Given we consider linear mixing, we are able to solve for the latent dimension $n$ by solving for the smallest integer $\hat{n}$ such that there exists a full column rank matrix $\tilde{H} \in \mathbb{R}^{d \times \hat{n}}$ where $\tilde{Z}= \tilde{H}^{\dagger} X$ has an open support in $\mathbb{R}^{\hat{n}}$, similar to Lemma 1 in [6]. We will add this information to Section 2 for clarity.

“the key part of identifiability theory seems to rely on the recent technique of score-based causal discovery method. In your setting, the latent factors themselves are identifiable before the linear mixing. I am wondering if other forms identifiable SCMs for the latent variables can also be identified?”

In considering other works that utilize score-matching to identify SCMs in the fully observable setting, we believe that our theoretical result could be extended to learn the upstream layer representations of nonlinear additive models with generic noise as an extension of [7], by modifying the principal to achieve identifiability in Eq (1) of our Lemma 3 to accommodate generic noise. The practical algorithm needs to be adapted accordingly as well. We will add this discussion to the paper.

“Regarding Proposition 1.: while a counter example is sufficient for this theorem, I'd like to ask if this failure case can be avoided by putting additional assumptions? e.g., …”

We appreciate this question regarding conditions for further identifiability. With additional faithfulness assumptions on the data generating process, we can potentially further identify the latent variables beyond "up to upstream layers". For example, assuming a generalized notion of faithfulness and access to soft interventions, [6] demonstrates that the underlying causal graph can be identified up to transitive closure. The mentioned condition by the reviewer can serve as one such faithfulness assumption for the 2-variable scenario. However, it is currently unclear what minimum faithfulness assumptions are required to achieve stronger identifiability guarantees in general scenarios, which we view as an important question for future work.

“typo: line 26: extend -> extent”

Thank you for identifying this. We will revise accordingly.

References:

[1] Nonlinear causal discovery with additive noise models
[2] Causal discovery with continuous additive noise models
[3] Score matching enables causal discovery of nonlinear additive noise models
[4] CAM: Causal additive models, high-dimensional order search and penalized regression
[5] Estimation of genetic networks and functional structures between genes by using Bayesian networks and nonparametric regression
[6] Identifiability guarantees for causal disentanglement from soft interventions
[7] Causal discovery with score matching on additive models with arbitrary noise
[8] Interventional causal representation learning

Thank you for recognizing the strength of our theoretical analysis and our proposed method! We would like to address your comments below:

“The method proposed in this paper relaxes some assumptions, making it potentially more applicable to real-world scenarios. It would be better to provide the performance of applying the proposed method to real-world data.”

We thank the reviewer for this suggestion. Similar to [1,2,3], we view this paper as having a primarily theoretical contribution, where our experiments on synthetic data provide a proof-of-concept for our main results. While we acknowledge the importance of real-world experiments, a major challenge lies in the fact that ground truth latent variables are not specifically labeled in many real-world datasets, making it difficult to evaluate the precise accuracy of estimated latent variables. Similar constraints are present in many previous works in causal disentanglement, where evaluations are generally based on synthetic data.

However, we do believe there are many real-world scenarios, such as topic modeling in natural language for example, that could be interesting settings to evaluate our methods. In particular, the method can be used to identify hierarchical topics at different layers of the underlying causal structure. Another example is learning a latent genealogical tree, where each layer representation would contain all of the prior ancestral information used to determine a given generation's traits. Such experiments would however necessitate an extensive amount of additional thought in experimental set up, which we believe is out of the scope of our current work, which mainly focuses on theoretical guarantees. We however recognize the importance of real-world applications and will add these examples to the introduction to build intuition of how our work might be useful beyond the theoretical guarantees.

In line 475, the denominator of the first term after the third equals sign in the equation should be differentiated with respect to $z_l$ instead of $k_l$?

Thank you for pointing this out. This term should be differentiated with respect to $z_l$ instead of $k_l$. We will edit accordingly.

References:

[1] General identifiability and achievability for causal representation learning
[2] Learning latent causal graphs via mixture oracles
[3] Learning linear causal representations from Interventions under general nonlinear mixing