Automatic Visual Instrumental Variable Learning for Confounding-Resistant Domain Generalization

We sincerely appreciate your support of the novelty of our work and your valuable comments. Here, we clarify and address these issues accordingly.

Q1: Ablation study:

R1: (1) Ablation studies on the three mutual information loss: We supplement ablation studies on the three mutual information loss terms (relevance, independence, and exclusion) for both versions of our model, VIV-DG and VIV-DG-Lite, on the PACS dataset, as shown below.

The results show that the exclusion loss is the most critical, as it effectively prevents the introduction of extra confounders as IVs. Meanwhile, the relevance and independence losses help reduce confounding effects and enhance the expressiveness of causal representations. These results, together with those from other datasets, will be included in the revised version.

Tab. 1 Ablation study of VIV-DG on PACS (ResNet18)

Tab. 2 Ablation study of VIV-DG-Lite on PACS (ResNet18)

(2) Ablation analysis of the regression predictor: The ablation analysis of the regression module is essentially reflected in the comparison between VIV-DG and VIV-DG-Lite: VIV-DG incorporates a regressor to predict causal representations, while VIV-DG-Lite does not. As shown in Table 1 of the paper, the performance of VIV-DG and VIV-DG-Lite varies across target domains. We provide further analysis to explain this: Overall, the regression predictor effectively prevents the learned causal representations from being affected by confounders.

Specifically, VIV-DG-Lite approximates the causal representations by combining the instrumental variables with the initial causal representations, which retains more original discriminative information but is more susceptible to confounding. Therefore, it performs better in domains with smaller distribution shifts, such as Product in Office-Home and Cartoon in PACS dataset.

In contrast, VIV-DG predicts causal representations through a trained regressor for bias correction, which helps reduce confounding effects and thus performs better in domains with larger distribution shifts, such as Sketch in PACS dataset. Overall, VIV-DG demonstrates greater robustness compared to VIV-DG-Lite.

Q2: Missing experimental results on other common backbones.

R2: The baseline methods we compare with mainly use ResNet18 and ResNet50. For fair comparison, we adopt the same backbones and do not conduct experiments on Vision Transformer. In fact, we use three different backbones in total: ResNet18, ResNet50, and a lightweight backbone with the same architecture as existing works on the Digits-DG dataset.

In addition, Lemma 2 and the Theorem provided in our response to Question 4 support that our method can automatically learn instrumental variables across different backbone architectures.

Q3: Experiments on large-scale datasets.

R3: We supplement experiments on the very large-scale domain generalization dataset DomainNet (586,575 images, 345 classes, and 6 domains). Due to the dataset’s size and long training time, we currently report results with Sketch as the target domain where distribution shift is large, and the other five domains as source domains. The results are as follows:

Tab. 3 Results on DomainNet (ResNet50)

Q4: Theoretical proofs of the VIV-DG.

R4: We supplement a theoretical analysis to demonstrate both the learnability and identifiability of Visual IVs, thereby establishing the effectiveness and theoretical soundness of our VIV-DG approach. Below we summarize the main theoretical results and proof ideas; full details will be included in the revised version:

• Lemma 1 (Identifiability Conditions for Visual IVs via (Conditional) Mutual Information): Let $(S, U, Y)$ be three random variables, where $S$ denotes the causal factor affecting $Y$, $U$ is the confounder, and $Y$ is the downstream label. A candidate variable $Z$ is called a visual instrument variable if it simultaneously satisfies: (i) Relevance: $I(Z;S)>0$; (ii) Independence: $I(Z;U)=0$; (iii) Exclusion: $I(Z;Y\mid S)=0$. Here $I(\cdot;\cdot)$ and $I(\cdot;\cdot\mid\cdot)$ denote (conditional) mutual information.

Proof idea: Based on the integral definition of mutual information, we establish the connection between IV conditions and mutual information. Each condition is verified individually to show that the three requirements for IVs can be rigorously expressed in terms of mutual information. We demonstrate that the three classical IV conditions—relevance, independence, and exclusion—can be rigorously characterized using (conditional) mutual information.

• Lemma 2 (Learnability of Visual IVs): Let $h_\psi:\mathrm{Image}\to Z$ be a parameterized mapping (e.g., via a deep network), with the following training objective

$

\mathcal{L}(\psi) =
-\alpha_1 I(Z;S\bigr) +\alpha_2 I(Z;U\bigr) +\alpha_3 I(Z;Y\mid S\bigr),

$

where $\alpha_1,\alpha_2, \alpha_3 >0$. Then any global minimizer $\psi^*$ satisfies

$I(h_{\psi^*}(\mathrm{Image});S)>0$,

$I(h_{\psi^*}(\mathrm{Image});U)>0$,

$I(h_{\psi^*}(\mathrm{Image});\,Y\mid S)>0$,
i.e. it exactly recovers a Visual IVs.

Proof idea: We define an objective function that combines the three mutual information terms, and prove that any global minimizer must satisfy all three conditions, thereby yielding a valid Visual IV. Since $-I(Z;S)\ge -\sup_\psi I(Z;S)$, $I(Z;U)\ge0$, $I(Z;Y\mid S)\ge0$, any global minimizer $\psi^*$ must simultaneously (i) maximize $I(Z;S)$, (ii) drive $I(Z;U)$ to zero, and (iii) drive $I(Z;Y\mid S)$ to zero; otherwise one could reduce $\mathcal{L}$ further. By the zero‐mutual‐information conditions we get $Z\perp\ U$ and $Z\perp\ Y\mid S$, and by the maximization $I(Z;S)>0$. These three facts are exactly the three requirements of the definition of Visual IVs.

• Theorem (Identifiability for Visual IVs): Let $(S, U, Y)$ be three random variables, where $S$ denotes the causal factor affecting $Y$, $U$ is the confounder, and $Y$ is the downstream label. Assume that $\mathcal H_{Z}=\{h_\psi:\mathrm{Image}\to Z\}$ is a sufficiently expressive family of mappings (e.g., one that contains all smooth bijections on a latent subspace). The optimize the objective

$

\mathcal{L}(\psi) =
-\alpha_1 I(h_\psi(\mathrm{Image});S\bigr) +\alpha_2 I(h_\psi(\mathrm{Image});U\bigr) +\alpha_3 I(h_\psi(\mathrm{Image});Y\mid S\bigr),

$

where $\alpha_1,\alpha_2, \alpha_3 >0$. Then any global minimizer $\psi^*$ yields

$

Z^* = h_\psi(\mathrm{Image})

$

which is identifiable in accordance with the definition of Vsual IV we defined.

Proof idea: We further prove from the perspectives of existence and uniqueness that under these mutual information constraints, the learned visual IV is identifiable in the function space and guaranteed to be unique under invertible transformations.

In addition to the theoretical support provided by the above theorems, we will also include a theoretical analysis of generalization performance in the revised version.

Q5: Experiments and analysis of the VIV-DG 's efficiency.

R5: Under the experimental settings described in the appendix, we supplement the analysis and experiments to evaluate the efficiency of VIV-DG, including convergence behavior, computational complexity, and GPU memory usage.

• Convergence Behavior Experiment: We record the convergence behavior during training and plot the corresponding convergence curves. The results show that VIV-DG converges quickly in the initial stage of causal representation learning. In the fourth stage (bias correction), the model consistently improves upon the accuracy achieved in the first stage. Additionally, in the second and third stages, the loss decreases rapidly, indicating stable and efficient training across all phases.

• Computational Complexity: The overall time complexity of our training framework is: $T = \mathcal{O}(N \cdot E \cdot P_{\max})$ where $N$ denotes the number of training samples, $E$ is the total number of training epochs, and $P_{\max}$ represents the maximum computational cost among the modules trained in each stage. This complexity does not involve exponential or polynomial growth, making it computationally tractable.

• GPU Memory Usage: Using ResNet-50 as the backbone, single-GPU training on an A800 GPU requires approximately 40+ GB of memory and runs stably without memory overflow or interruption.

The primary computational cost arises from the automatic visual IV learning module. However, we consider this a reasonable and acceptable trade-off, given the improvements in automation and generalization capability that it brings.

We are grateful for your recognition of the theoretical effectiveness and conceptual interest of our proposed visual IV learning framework. Here, we clarify and address these issues accordingly.

Questions:

Q1: Why is VIV-DG not included in Table 2?

R1: We thank the reviewer for pointing this out. This was our oversight. We now supplement the results of VIV-DG on the VLCS dataset, further demonstrating the effectiveness of VIV-DG.

Tab. 1 Results of VIV-DG on VLCS

Q2: Positive and negative samples. Confounder indices.

R1: (1) Explanation of positive and negative samples generated. In Section 3.4.2, positive sample pairs consist of visual instrumental variables $Z$ and causal representations $S$ from the same original sample, denoted as ${(z_i, s_i)}_{i=1}^N$, where both $z_i$ and $s_i$ originate from the image $x_i$. Negative sample pairs are formed by combining $Z$ and $S$ from different samples, $(z_i, s_j)$, where $i \neq j$, created by cross-sample indexing. Similarly, in Section 3.4.3, positive sample pairs consist of $Z$ and confounders $U$ from the same image, while negative sample pairs combine $Z$ with shuffled confounders $U$ from different samples. This construction of positive and negative samples effectively approximates mutual information estimation during mini-batch training.

(2) Explanation of confounder indices: The indices of confounders are automatically identified and separated by the model through the positive-negative sample contrastive mechanism during training, rather than being explicitly annotated or manually defined.

Weaknesses:

W1: Explanation of the rationality of the causal graph.

R1: We apologize for the misunderstanding caused by our unclear explanation and appreciate the opportunity to clarify. In Fig. 4, the unobserved confounder $C$ does not have a direct causal link to the input image $X$, which is an intentional modeling choice. Without loss of generality, we assume that $C$ represents confounders related to unknown domain styles or backgrounds. Since domains cannot be exhaustively enumerated, such unobserved confounders inevitably exist. However, these factors do not appear in the current observed input data or known source domains, and therefore do not exert a direct causal effect on the input data $X$. Conversely, adding a direct edge from $C$ to $X$ would introduce ambiguity by implying that $C$ is the observed confounder.

It is worth noting that both known and unknown domains share the same causal features $S$ and potential outcome $Y$. In the unseen domain where the confounder $C$ exists, $C$ also affects $S$ and $Y$. Therefore, we model this by adding causal edges from $C$ to both $S$ and $Y$ in Fig. 4.

W2: Not evaluated on DomainBed and comparisons with other causal methods.

R1: (1) Standard DomainBed benchmark: Although we do not run our experiments directly on the DomainBed codebase, our experimental setup strictly follows the standard DomainBed protocol, including dataset splits, backbones, and the leave-one-domain-out strategy. For fair comparison，our main extraction network (initial training + bias correction fine-tuning) is trained for 50 epochs, consistent with the compared methods.

(2) More causal methods: We further investigated and included several causal methods. The results are as follows. Our method consistently outperforms these approaches.

Tab. 2 Results on Digits-DG

Tab. 3 Results on PACS (ResNet18)

Tab. 4 Results on PACS (ResNet50)

Tab. 5 Results on VLCS (ResNet50)

[1] Unbiased Semantic Representation Learning Based on Causal Disentanglement for Domain Generalization. (2024)

[2] A Causality-Aware Perspective on Domain Generalization via Domain Intervention. (2024)

[3] Out-of-distribution Generalization with Causal Invariant Transformations. (2022)

W3: Time complexity.

R3: (1) Computational Complexity: The overall time complexity of our training framework is: $T = \mathcal{O}(N \cdot E \cdot P_{\max})$ where $N$ denotes the number of training samples, $E$ is the total number of training epochs, and $P_{\max}$ represents the maximum computational cost among the modules trained in each stage. This complexity does not involve exponential or polynomial growth, making it computationally tractable.

(2) The primary computational cost arises from the automatic visual IV learning module. However, we consider this a reasonable and acceptable trade-off, given the improvements in automation and generalization capability that it brings. Specifically, the training process of VIV-DG is slightly more complex and time-consuming than some baseline methods, mainly due to its multi-stage Visual IV learning. However, unlike predefined IV methods, our approach does not rely on domain experts for IV design and selection, avoiding high human costs and prior bias. It also offers better adaptability and scalability.

W4: More comprehensive examples.

R4: We appreciate the suggestion and will include more representative examples of confounders in the revised version, such as scene background, co-occurrence bias, image style, lighting conditions, and camera angles. They are typically entangled and difficult to disentangle in the absence of labeled supervision. These confounders tend to exhibit similar patterns in the Fourier amplitude information of images, such as color, style, and brightness. Therefore, we propose a feature extraction method based on reconstructing images from amplitude information to model the confounders. Our Visual IV method mitigates the influence of both observed and unobserved confounders without explicitly modeling them.

W5: The paper should clarify what can be visualized in the original figure or visualize the features learned for the IV and causal features.

R5: We apologize for the unclear explanation of the visualization experiments, and we clarify the details as follows. We compare our VIV-DG with CIRL, a strong causal domain generalization baseline, in the visualization experiments. We intentionally select the Art Painting domain—where the classification accuracies of the two methods are similar—to demonstrate that our method better captures causal regions. Even if the improvement is not visually striking, it still shows progress in identifying causal cues.

To further demonstrate the causal representation learning ability of our method, we extend the Grad-CAM visualizations to the Cartoon and Sketch domains. (We cannot display the images here.) The results further show that VIV-DG focuses more on the central structure and semantically relevant object regions, which is consistent with the superior performance of VIV-DG over CIRL in these domains.

Additional visualization evidence: To verify that the learned instrumental variables alleviate confounding effects, we visualize the Grad-CAM maps of causal features before and after IV-based correction. The visual results show that the model captures more robust and semantically meaningful regions after using the visual IVs.

Quantitative verification through mutual information: Since visual results cannot be displayed here, we also provide more direct and quantitative evidence of IV effectiveness. We compute the estimated mutual information values of $I(Z;S)$, $I(Z;U)$, and $I(Y;Z|S)$ to verify whether the learned representation $Z$ satisfies the relevance, independence, and exclusion conditions of an instrumental variable. The results indicate that the learned representation approximately satisfies the IV conditions and thus serves as an effective guidance for mitigating confounding bias. We will include these results in the revised version.

Tab. 6 Estimated values of mutual information on PACS

W6: Ablation studies.

R6: We supplement ablation studies on the three mutual information loss terms (relevance, independence, and exclusion) for both versions of our model, VIV-DG and VIV-DG-Lite, on the PACS dataset, as shown below.

Tab. 7 Ablation study of VIV-DG on PACS (ResNet18)

Tab. 8 Ablation study of VIV-DG-Lite on PACS (ResNet18)

The results show that the exclusion loss is the most critical, as it effectively prevents the introduction of extra confounders as instrumental variables. Meanwhile, the relevance and independence losses work together to reduce confounding effects and enhance the expressiveness of causal representations. These three loss terms collaboratively contribute to the strong performance of both the full and Lite versions. These results, together with those from other datasets, will be included in the revised version.

We sincerely appreciate your support of the novelty of our work and your valuable comments. Your suggestions help strengthen our paper. Here, we clarify and address these issues accordingly.

Questions:

Q1: Could the authors explain why the results vary in their experiments with both versions of the model and on how many iterations or different seeds the results were evaluated?

R1: (1) Analysis of result variations between the two versions: The performance differences between the two model versions mainly arise from their different strategies for predicting causal factors from Visual IVs.

VIV-DG-Lite approximates the causal factors by combining the instrumental variables with the initial causal features, which retains more original discriminative information but is more affected by confounders. Therefore, it performs better in domains with smaller distribution shifts, such as Product in Office-Home and Cartoon in PACS.

In contrast, VIV-DG predicts causal factors via a trained regressor for bias correction, which reduces confounding effects and thus performs better in domains with larger distribution shifts, such as Sketch in PACS. Overall, VIV-DG is more robust than VIV-DG-Lite.

(2) Explanation regarding how many iterations or different seeds the results were evaluated. We provide a brief report of the training epochs for VIV-DG and VIV-DG-Lite in Tables 4 and 5 of the appendix. Here, we offer a more detailed explanation for clarity: VIV-DG is trained for a total of 100 epochs, while VIV-DG-Lite is trained for 70 epochs in total, as it removes the regression stage compared to VIV-DG. Specifically, for fair comparison, we fix the number of training epochs of the main encoder (initial training + bias correction fine-tuning) to 50 for both models, which is consistent with the baseline methods. All reported results are averaged over 3 runs with different random seeds. We will clarify these settings in the revised version.

Q2: Are there any ablations done on why the Lite version of the model performs as good as the normal model on certain domains?

R2: We supplement ablation studies on the three mutual information loss terms (relevance, independence, and exclusion) for both versions of our model, VIV-DG and VIV-DG-Lite, on the PACS dataset, as shown below.

Tab. 1 Ablation study of VIV-DG on PACS (ResNet18)

Tab. 2 Ablation study of VIV-DG-Lite on PACS (ResNet18)

The results show that the exclusion loss is the most critical, as it effectively prevents the introduction of extra confounders as instrumental variables. Meanwhile, the relevance and independence losses work together to reduce confounding effects and enhance the expressiveness of causal representations. These three loss terms collaboratively contribute to the strong performance of both the full and Lite versions. These results, together with those from other datasets, will be included in the revised version.

The Lite version shares the same mutual information constraint components as the full version. Notably, the Lite version derives new causal factors by aggregating the instrumental variables with the initial causal representations, which retains more of the original discriminative information and enables it to achieve comparable performance to the full version in certain domains.

Q3: VRAM usage, Scalability, Training cost vs. performance.

R3: (1) It takes how much VRAM: By using ResNet-50 as the backbone, single-GPU training on an A800 GPU requires about 40+ GB of memory and runs stably.

(2) Scalability to larger backbones: We provide theoretical analyses of the identifiability and learnability of visual IVs (as shown below), showing that any extracted features that satisfy the IV conditions can serve as approximate visual IVs. This theoretical grounding ensures that our method remains valid regardless of the backbone architecture. As a result, VIV-DG scales effectively to larger backbones in domain generalization training while still producing reliable visual IVs.

• lemma[Learnability of Visual IVs] Let $h_\psi:\mathrm{Image}\to Z$ be a parameterized mapping (e.g., via a deep network), with the following training objective

$

\mathcal{L}(\psi) =
-\alpha_1 I(Z;S\bigr) +\alpha_2 I(Z;U\bigr) +\alpha_3 I(Z;Y\mid S\bigr),

$

where $\alpha_1,\alpha_2, \alpha_3 >0$. Then any global minimizer $\psi^*$ satisfies

$I(h_{\psi^*}(\mathrm{Image});S)>0$,

$I(h_{\psi^*}(\mathrm{Image});U)>0$,

$I(h_{\psi^*}(\mathrm{Image});\,Y\mid S)>0$,
i.e. it exactly recovers a Visual IVs.

Proof idea: Since $-I(Z;S)\ge -\sup_\psi I(Z;S)$, $I(Z;U)\ge0$, $I(Z;Y\mid S)\ge0$, any global minimizer $\psi^*$ must simultaneously (i) maximize $I(Z;S)$, (ii) drive $I(Z;U)$ to zero, and (iii) drive $I(Z;Y\mid S)$ to zero; otherwise one could reduce $\mathcal{L}$ further. By the zero‐mutual‐information conditions we get $Z\perp\ U$ and $Z\perp\ Y\mid S$, and by the maximization $I(Z;S)>0$. These three facts are exactly the three requirements of the definition of Visual IVs.

• Theorem (Identifiability for Visual IVs): Let $(S, U, Y)$ be three random variables, where $S$ denotes the causal factor affecting $Y$, $U$ is the confounder, and $Y$ is the downstream label. Assume that $\mathcal H_{Z}=\{h_\psi:\mathrm{Image}\to Z\}$ is a sufficiently expressive family of mappings (e.g., one that contains all smooth bijections on a latent subspace). The optimize the objective

$

\mathcal{L}(\psi) =
-\alpha_1 I(h_\psi(\mathrm{Image});S\bigr) +\alpha_2 I(h_\psi(\mathrm{Image});U\bigr) +\alpha_3 I(h_\psi(\mathrm{Image});Y\mid S\bigr),

$

where $\alpha_1,\alpha_2, \alpha_3 >0$. Then any global minimizer $\psi^*$ yields

$

Z^* = h_\psi(\mathrm{Image})

$

which is identifiable in accordance with the definition of Vsual IV we defined.

Proof idea: We further prove that, under these constraints, the learned visual IV is identifiable in the function space, with uniqueness guaranteed up to invertible transformations.

(3) Trade-off between training cost, interpretability, and automation:

We consider the computational cost of VIV-DG to be commensurate with its higher degree of automation, enhanced interpretability, and improved scalability. The main computational cost comes from the automatic Visual IV learning module. Although this introduces additional training cost, it significantly reduces the manual effort required to design instrumental variables and avoids reliance on domain knowledge or expert heuristics.

Importantly, we supplement our work with theoretical analysis proving that the automatically learned Visual IVs strictly satisfy the IV conditions, which enhances the interpretability of the model.

While the performance improvements over some baseline methods are moderate, our approach consistently outperforms predefined instrumental variable methods such as IV-DG in terms of accuracy and practical applicability. Therefore, we consider the computational cost a reasonable trade-off.

Moreover, the computational cost is also a limitation we mention in the paper. To address this, we provide a simplified version, VIV-DG-Lite, which removes one training stage, significantly reducing the computational cost while still maintaining competitive accuracy.

Weaknesses:

W1: The results seem a bit random. Authors cite that in PACS that the framework surpasses all baselines, when in actuality it is equal or lower in three out of four domains.

R1: (1) Explanation of Experimental Results: At first glance, the experimental results may seem less favorable on individual domains. However, it is important to note that our two versions, VIV-DG and VIV-DG-Lite, are also compared against each other. As shown in Table 1 of the paper, VIV-DG achieves the best performance only on the Photo domain in the PACS dataset, which may suggest limited advantage at first glance. However, a closer examination reveals that VIV-DG is mainly outperformed by its simplified version, VIV-DG-Lite, in the domains where it ranks second.

If VIV-DG-Lite is excluded from the comparison, VIV-DG achieves the best average performance on PACS, ranks first in two domains, and second in the remaining two.

Furthermore, beyond PACS, VIV-DG achieves the best results on two domains of Digits-DG with a notable improvement in average performance, and outperforms all baselines on three out of four domains in Office-Home.

W2: Lack of explanation for why the Lite version outperforms in some domains, and missing results of the normal version on VLCS.

R2: (1) Analysis of result variations between the two versions: The performance differences and detailed analysis between VIV-DG-Lite and VIV-DG across domains are provided in our response to Question 1.

(2) Results of normal version VIV-DG on VLCS: We thank the reviewer for pointing this out. This was our oversight. We supplement the results of VIV-DG on the VLCS dataset, as shown below, further demonstrating the effectiveness of VIV-DG. We will incorporate these results into Table 2 in the revised version.

Tab. 3 Results of VIV-DG on VLCS

W3: Marginal gains vs. Computational cost.

R3: We provide a detailed explanation; please refer to our response to Question 3.

We sincerely appreciate your support of the novelty of our work and your valuable comments. The suggestions help improve our work, and we clarify and address the raised concerns as follows.

Questions:

Q1: Direct verification that the learned IV representation satisfies the IV conditions.

R1: We provide validation from both experimental and theoretical perspectives:

(1) Experiments on the estimated values of mutual information: We conduct additional experiments to estimate and quantify the mutual information terms $I(Z;S)$, $I(Z;U)$ and $I(Y;Z|S)$. The results indicate that the learned Visual IV satisfies the relevance, independence, and exclusion conditions, confirming its validity.

Tab. 1 Estimated values of mutual information on PACS

(2) Ablation studies on the three mutual information loss: We supplement ablation studies on the three mutual information loss terms (relevance, independence, and exclusion) for both versions of our model on the PACS dataset, as shown below. The results show that the exclusion loss is the most critical, as it effectively prevents the introduction of extra confounders as IVs. Meanwhile, the relevance and independence losses help reduce confounding effects and enhance the expressiveness of causal representations. These results, together with those from other datasets, will be included in the revised version.

Tab. 2 Ablation study of VIV-DG on PACS (ResNet18)

Tab. 3 Ablation study of VIV-DG-Lite on PACS (ResNet18)

(3) Theoretical Support:

We supplement a theoretical analysis to support the learnability and identifiability of Visual IVs. Below we summarize the main theoretical results and proof ideas; full details will be included in the revised version:

• Lemma (Learnability of Visual IVs): Let $h_\psi:\mathrm{Image}\to Z$ be a parameterized mapping (e.g., via a deep network), with the following training objective

$

\mathcal{L}(\psi) =
-\alpha_1 I(Z;S\bigr) +\alpha_2 I(Z;U\bigr) +\alpha_3 I(Z;Y\mid S\bigr),

$

where $\alpha_1,\alpha_2, \alpha_3 >0$. Then any global minimizer $\psi^*$ satisfies

$I(h_{\psi^*}(\mathrm{Image});S)>0$,

$I(h_{\psi^*}(\mathrm{Image});U)>0$,

$I(h_{\psi^*}(\mathrm{Image});\,Y\mid S)>0$,
i.e. it exactly recovers a Visual IV.

Proof idea: We define an objective function that combines the three mutual information terms, and prove that any global minimizer must satisfy all three conditions, thereby yielding a valid Visual IV.

• Theorem (Identifiability for Visual IVs): Let $(S, U, Y)$ be three random variables, where $S$ denotes the causal factor affecting $Y$, $U$ is the confounder, and $Y$ is the downstream label. Assume that $\mathcal H_{Z}=\{h_\psi:\mathrm{Image}\to Z\}$ is a sufficiently expressive family of mappings (e.g., one that contains all smooth bijections on a latent subspace). The optimize the objective

$

\mathcal{L}(\psi) =
-\alpha_1 I(h_\psi(\mathrm{Image});S\bigr) +\alpha_2 I(h_\psi(\mathrm{Image});U\bigr) +\alpha_3 I(h_\psi(\mathrm{Image});Y\mid S\bigr),

$

where $\alpha_1,\alpha_2, \alpha_3 >0$. Then any global minimizer $\psi^*$ yields

$

Z^* = h_\psi(\mathrm{Image})

$

which is identifiable in accordance with the definition of Vsual IV we defined.

Proof idea: We further prove that, under these constraints, the learned visual IV is identifiable in the function space, with uniqueness guaranteed up to invertible transformations.

Q2: Explanation of Visualizations and Toy Example.

R2: (1) Explanation for the Grad-CAM visualizations: We apologize for the unclear explanation of the visualization experiments, and we clarify the details as follows. We compare our VIV-DG with CIRL, a strong causal domain generalization baseline, in the visualization experiments. We intentionally select the Art Painting domain—where the classification accuracies of the two methods are similar—to demonstrate that our method better captures causal regions. Even if the improvement is not visually striking, it still shows progress in identifying causal cues.

To further demonstrate the causal representation learning ability of VIV-DG, we extend the Grad-CAM visualizations to the Cartoon and Sketch domains. (We cannot display the images here.) The results further show that VIV-DG focuses more on the central structure and semantically relevant object regions, which is consistent with the superior performance of VIV-DG over CIRL in these domains.

Additional visualization evidence: To verify that the learned IVs alleviate confounding effects, we visualize the Grad-CAM maps of causal features before and after IV-based correction. The results show that the model captures more robust and semantically meaningful regions after using the visual IVs.

(2) Toy example: Consider an image of a dog on the grass, where the dog represents the causal feature $S$ and the grass serves as the confounder $U$. Methods like CIRL aim to disentangle causal features but do not explicitly remove confounding bias, so the learned representations may still retain spurious signals (e.g., grass color). In contrast, our method learns a visual IV $Z$ that satisfies the IV conditions and uses it to predict a new causal feature $\hat{S}$ via a regressor, which is unbiased by confounders. Since $Z$ is explicitly independent of $U$ and influences $Y$ only through $S$, the predicted $\hat{S}$ is free from confounding interference. It is determined solely by $Z$ and guided by supervision from $Y$, which encourages $\hat{S}$ to capture mainly causal information. This results in more robust and accurate causal representations.

Weaknesses:

W1: Robustness analysis of the masking process.

R1: We add a sensitivity experiment on the Digits-DG dataset using threshold δ ranging from 20 to 40, as shown below, and observe minimal performance variation, indicating good robustness.

Tab. 4 Sensitivity to δ on Digits-DG

We use a grayscale difference δ of 30 to generate the mask, which is a common empirical value in image processing. This simple strategy helps reduce computational cost while maintaining efficiency. In future work, we plan to develop an automated mask generation mechanism to further enhance robustness and automation.

W2: Lack of comparison with IV-DG on the Digits-DG dataset and analysis of their respective advantages and disadvantages.

R2: (1) Explanation for not including a comparison with IV-DG on Digits-DG: Following common practice, we report IV-DG results directly from the original paper for fair comparison. IV-DG is only evaluated on PACS and Office-Home in its original paper, and not on other benchmarks such as Digits-DG. Therefore, we do not include the results of IV-DG on the Digits-DG benchmark. Notably, compared to IV-DG, our method achieves improvements of +2.6% on PACS and +3.0% on Office-Home, demonstrating a clear advantage.

(2) Learnable IVs vs. Predefined IVs: Regarding the limitations of predefined IVs: Classical IV assumptions—especially the exclusion condition—are strict and often difficult to verify in practice. For instance, IV-DG selects data from one domain as the IV for another, which can easily violate the exclusion condition. Consider using a cartoon image of a dog as the IV for an art-style image of the same dog. Although the domains differ, the semantic label is the same, and features extracted from the cartoon image may directly predict the label "dog", thus violating the exclusion requirement and leading to weak or invalid IVs. In contrast, our learnable IV approach explicitly enforces the IV conditions during training, thereby avoiding such violations and yielding more reliable causal representations.

W3: The performance improvements are relatively modest, the model introduces additional complexity, and the Grad-CAM visualizations do not clearly illustrate whether more causal representations are being learned.

R3: (1) Computational cost. We consider the computational cost of VIV-DG to be a reasonable trade-off for achieving both automation and scalability. The main computational cost of VIV-DG comes from the automatic visual IV learning module. Although this introduces additional training cost, it substantially reduces the manual effort required to design IVs, which is often necessary in predefined IV-based methods. It also avoids reliance on domain knowledge or expert heuristics. Despite achieving only moderate gains over some baselines, it consistently outperforms predefined IV-based methods like IV-DG in both performance and practicality.

Moreover, the computational cost is also a limitation we mention in the paper. To address this, we provide a simplified version, VIV-DG-Lite, which removes one training stage, significantly reducing the computational cost while still maintaining competitive accuracy.

(2) Theoretical Support: We provide a theoretical analysis of the identifiability of visual IV, please see our response to Question 1. This further supports the learnability and validity of our approach.

(3) Grad-CAM visualizations: For clarification and analysis of the Grad-CAM visualizations, please see our response to Question 2.