W1: Motivation for having two-level representations.

Response: Thanks for the comment!

We first emphasize that existing invariant or causal representation learning methods typically focus on learning a single-layer representation mapping $\phi: X \rightarrow Z$. While effective in some settings, these approaches face several key challenges in the context of anti-causal learning, as outlined in our introduction:

• They often assume perfect interventions, which are difficult or impractical to implement in real-world scenarios.
• They typically require access to explicit structural dependencies (e.g., via Structural Causal Models), which are rarely available.
• Many methods also assume IID data or require knowledge of the test-time distribution, limiting their applicability in realistic settings involving distribution shift.

We propose to address all the challegnes through the two-level causal hierarchy framework.

• Low-level representation mapping ($\phi_L: X \rightarrow Z_L$) extracts features directly from raw observations (without requiring explicit DAGs), capturing the anti-causal structure of the data, including both causal ($Y \rightarrow X$) and spurious ($E \rightarrow X$) patterns. The learned low-level representations support reasoning under both perfect and imperfect interventions, enabled by the introduced interventional kernels.
• High-level representation mapping ($\phi_H: Z_L \rightarrow Z_H$) is applied on top of $\phi_L$. It serves as an information bottleneck that extracts causally invariant features — specifically, preserving the effect of $Y \rightarrow X$ while discarding spurious environment-related signals from $E \rightarrow X$.
• Together, the two-level mappings cooperate to learn robust anti-causal representations, with $\phi_L$ capturing rich generative mechanisms and $\phi_H$ filtering out non-causal noise for improved generalization across environments.

Q1/W2: When to apply the proposed method/How to know anti-causal relationship in general.

Response: An anti-causal relationship occurs when the task label causes the observational features. This structure is common in many real-world applications. For instance, we show some examples below:

• Medical diagnosis: The presence of a disease causes specific symptoms or test results.
• Image classification: The identity of an object determines its visual appearance.
• Quality control: Defects in a product lead to anomalies in sensor measurements.
• Recommendation system: Users' preferences drive their behavior patterns.

Technically, we can identify anti-causal relationships through interventional tests: if intervening on features X does not change the label Y, but intervening on Y does affect X, then the relationship is anti-causal. Our Corollary 2 (Interventional Kernel Invariance) formalizes this intuition and provides a principled test using the interventional kernel we introduced.

Q2: Difference between "Invariant Across Environments" and "Environment-Independent"

Response: Thank you for the comment, and we apologize for the confusion.

• Invariant across environments refers to consistency under distribution shifts. Specifically, it means that the learned representation remains stable across different environments/distributions.
• Environment-independent, on the other hand, implies that the representation is statistically independent of the environment variable. That is, the features do not contain any information about the environment.

Mapping this distinction to our two-layer representation learning setup:

• The low-level representation $\phi_L$ learns features such that the distribution $\Pr(\phi_L(\mathcal{X}) \mid Y)$ is invariant across environments, but it may still encode residual information about the environment $E$.
• The high-level representation $\phi_H$ then filters out this residual information to enforce conditional environment-independence, i.e., $\phi_H(\phi_L(\mathcal{X})) \perp E \mid Y$.

Illustrative example: Two environments may exhibit identical $\Pr(\phi_L(\mathcal{X}) \mid Y)$, meaning the representations appear invariant across environments. However, the features $\phi_L(\mathcal{X})$ may still contain environment-specific markers (e.g., background texture or domain-specific artifacts). $\phi_H$ further removes these markers to achieve true environment-independence.

We will revise the wording in the paper to better reflect this distinction.

Q3: How does Causal Dynamic learn low-level representations? How is it different from Causal Abstraction? How are high-level and low-level feature parts determined in a neural network?

Response: Thank you for the valuable question!

Recall that the low-level representation mapping $\phi_L$ aims to capture the anti-causal structure, i.e., both $Y \rightarrow X$ and $E \rightarrow X$. The Causal Dynamic module facilitates this learning by optimizing a supervised loss in conjunction with the $R_2$ regularizer. Minimizing $R_2$ encourages the low-level representation to align with the true causal mechanisms underlying the data, promoting better generalization under intervention.

Building on top of $\phi_L$, the Causal Abstraction module learns a high-level representation mapping $\phi_H$, guided by the $R_1$ regularizer. The goal is to further enforce statistical independence from the environment variable $E$. Specifically, $R_1$ measures the discrepancy between the expected high-level representations across environments, conditioned on the label $Y$, and minimizing it removes environment-specific information while retaining causal features.

In practice, we design the architecture with the following principles:

• The low-level representation should be sufficiently rich to capture both label-relevant and environment-dependent signals (i.e., $Y \rightarrow X$ and $E \rightarrow X$).
• The high-level representation serves as an information bottleneck, filtering out environment-dependent variations, and thus is set to a lower dimension than the low-level representation.

In our experiments with image data:

• $\phi_L$ is implemented as a CNN encoder, producing a $d_L$-dimensional representation.
• $\phi_H$ is a small MLP bottleneck, outputting a $d_H$-dimensional high-level representation, where $d_H < d_L$.

L1: Discuss the limitations of the proposed work

Response: Thanks for the suggestion! We outline the potential limitations below:

• Scalability: Generalizing ACIA to large-scale, high-dimensional data may be computationally expensive and time-consuming.
• Optimal regularization selection: The optimal hyperparameters $\lambda_1$ and $\lambda_1$ in the objective function may dependent on the application domain. It requires careful tuning to obtain their optimal values.
• Complex causal structures: ACIA is primarily designed for anti-causal settings. Its applicability to more complex causal structures—such as confounded-descendant or confounded-outcome [59] scenarios—remains unclear and warrants further investigation.

Dear reviewer qtK8,

I'd like to invite you to actively engage in the discussion with the authors. They provided a detailed rebuttal to your review, and it is important to know if this new information changes your mind.

Best regards, AC

Dear Reviewer qtK8,

Thank you for the reply — we’re glad that our previous response has addressed all comments except for the distinction between “Invariant across environments” and “Environment-independent.”

We apologize for the confusion, which we believe primarily stems from the imprecise use of the term “invariant.”

As noted in our earlier rebuttal:

“Low-level representation learns features that are invariant across environments, which refers to consistency/stability under distribution/environment shifts.”

This statement was not intended to imply that we achieve

P(\Phi_L(X) \mid Y, E) = P(\Phi_L(X) \mid Y), $$ but rather that the learned conditional distribution $P(\Phi_L(X) \mid Y)$ is **stable with respect to environment shifts**, given that the label prior $P_e(Y)$ may vary across environments. In this context, we meant that the **low-level features may still encode residual information about the environment** (i.e., allow predicting $E$ from $\phi_L(X)$ when $Y$ is known), particularly due to the differences in $P_e(Y)$ across environments. Moreover, we emphasize that in the **anti-causal setting**, it is **impossible to learn a single-layer representation** $\Phi_L$ that satisfies $P(\Phi_L(X) \mid Y, E) = P(\Phi_L(X) \mid Y)$, as demonstrated in Ref[59]. This theoretical limitation further motivates our introduction of **high-level representation learning**, which aims to explicitly remove the influence of the environment. Or to say, forcing the **environment independence** in the high-level representation explicitly reflected by the conditional independence: $ \phi_H(\phi_L(X)) \perp E \mid Y. $ To prevent future confusion, we will revise the terminology and replace *“invariant”* with more precise phrases such as *“stable across environments”* or *“stable under environment shift.”* **We hope this clarification has addressed your last remaining concern—thank you again for your thoughtful feedback!** Authors

Q1: Visualize causal relationship between different features from the two-level representation learning.

Response: Thank you for the valuable suggestion!

To visualize the causal relationship between low-level and high-level features, we will employ gradient-based attribution methods (visualization via heatmap) to quantify how each dimension in the low-level representation contributes to each dimension in the high-level representation. For instance, on CMNIST, this approach will reveal how low-level features—such as digit shape and color—are transformed through the two-stage representation learning process. Specifically, we will generate flow diagrams illustrating:

• Which low-level neurons are activated by shape vs. color,
• And how only shape-related dimensions in low-level representations are used by high-level representations to form causally invariant features (i.e., digit identity).

This visualization will help confirm that the high-level representation successfully filters out environment-specific information while preserving task-relevant signals.

Q2: Explicit DAG is abandoned during the optimization. Comparison of the prior information used in different methods.

Response: We highlight that eliminating dependency on explicit DAG is actually one of the key advantages of ACIA. This is due to the fact that obtaining the accurate and explicit DAG in real-world scenarios is often difficult and becomes our motivation.

Please see below table the prior information of the compared SOTA causal representation learning methods. Our experimental results in the paper demonstrate that, though requiring less information, ACIA still outperforms the compared SOTAs.

Note:

• "Variable roles only" refers to knowing the role of each variable in the causal model, i.e., identifying which variable is the target (Y), which are observations (X), and which represent environmental factors (E).
• "Variable types" means knowing the data type of each variable, such as whether it is binary, categorical, or drawn from a specific noise distribution.
• "Variable relationship" refers to knowing the causal structure (i.e., the directionality in the causal DAG) among the variables.

Q3: Provide the theoretical analysis of the intervention assumption.

Response: Perfect and imperfect intervention types are unified in the the introdued interventional kernel framework, stated below:

Theorem 2 (Interventional Kernel): For any intervention $\mathbb{Q}: \mathscr{H} \times \Omega \rightarrow [0,1]$, there exists a unique interventional kernel:

$K_S^{do(\mathcal{X}, \mathbb{Q})}(\omega, A) = \int_{\Omega} K_S(\omega, d\omega') \mathbb{Q}(A|\omega')$

In perfect interventions, $\mathbb{Q}(A|\omega') = \mathbb{Q}(A)$, while $\mathbb{Q}(A|\omega')$ varies with $\omega'$ in imperfect interventions.

Further, Corollary 2 shows that interventional kernel invariance properties $K_S^{do(X)}(\omega, \{Y \in B\}) = K_S(\omega, \{Y \in B\})$ hold regardless of intervention perfectness.

W1: More details on the two-level representation algorithm

Response: Due to space constraints, we have provided the full details of the two-level representation algorithm in Algorithms 1–3 (Pages 44 and 55) in Appendix. Do these pseudocode descriptions help clarify the method?

W2/L1: Nessisity of the min-max adversarial training approach.

Response: ACIA adopts a min-max optimization framework to ensure worst-case robustness across environments. This formulation is theoretically grounded in our OOD generalization bounds (Theorem 7), which require guarantees on worst-case performance across diverse environments.

We emphasize that existing (environment-)invariant representation learning methods—such as IRM, Rex, and VRex—are designed to optimize for worst-case performance across environments. This worst-case objective is critical: without it, the learned representations often fail to disentangle environmental factors effectively, as has been extensively validated in prior literature.

Dear reviewer wvHG,

I'd like to invite you to actively engage in the discussion with the authors. They provided a detailed rebuttal to your review, and it is important to know if this new information changes your mind.

Best regards, AC

Q1: Clarify the roles of the two-level learning

Response: The two-level representation learning process in our framework serves distinct roles within the causal hierarchy. We provide the following clarification:

• Low-level representation mapping ($\phi_L$) aims to extract features directly from raw observations (without requiring explicit DAGs), capturing the anti-causal structure of the data. This includes how both the label variable $Y$ and the environment variable $E$ influence the observation $X$. The learned features support reasoning under both perfect and imperfect interventions, enabled by the introduced interventional kernels.
• High-level representation mapping ($\phi_H$) is applied on top of $\phi_L$, and its purpose is to extract causally invariant features — specifically, preserving the effect of $Y \rightarrow X$ while discarding spurious environment-related signals from $E \rightarrow X$. Formally, it enforces the condition: $\phi_H(\phi_L(\mathcal{X})) \perp E \mid Y$, ensuring that the final representation is invariant to environment-specific variations.

Together, the two-level mappings cooperate to learn robust anti-causal representations, with $\phi_L$ capturing rich generative mechanisms and $\phi_H$ filtering out non-causal noise for improved generalization across environments.

Q2: Figure 1 and Figure 2 mergerd into one

Response: Thanks for the suggestion! We will merge the two figures!

Q3: Typos: Line 194 "Appendix ??"; “objection” -> “objective”; "P(Y|do(X=x)" -> "P(Y|do(X=x))"

Response: Thanks for pointing them out and correction!

Q1/W1: Empirical training time on the evaluated datasets

Response: The table below reports the runtime on a laptop with a single GPU (3.3 GHz, 32 GB RAM), using a desired precision $\epsilon=0.01$ and failure probability $\delta=0.05$. We observe that ACIA completes in approximately 3–16 minutes under these settings.

Q2: Appendix Placeholder in Line 194

Response: Thanks for pointing out. It is Appendix C.5.