The most fundamental weakness lies in the theory. The proofs of Theorem 1 and 2 are incorrect. Specifically:

1. Equal in distribution does not mean two random variables are equal. The first step of the proof for theorem 1 is problematic.
2. Similarly, the proof of theorem 2 is incorrect because we cannot deduce Z = M\tilde{Z} from their probability distributions.
3. I don’t think the current proof can be made correct. In general, the linear identifiability requires supervision [1], and the permutation identifiability requires additional constraints on the generative process [2]. It is unclear how the setting considered here is relevant with these.
4. Without permutation identifiability, the inferred latent factors are actually arbitrarily entangled. The linear identifiability is insufficient for the use case.

At this point, I don’t think additional discussions are needed.

[1] Khemakhem, Ilyes, et al. "Variational autoencoders and nonlinear ica: A unifying framework." International conference on artificial intelligence and statistics. PMLR, 2020.

[2] Lachapelle, Sébastien, et al. "Disentanglement via mechanism sparsity regularization: A new principle for nonlinear ICA." Conference on Causal Learning and Reasoning. PMLR, 2022.

1.The authors propose the “Spatial Factor” for modeling spatial dependencies, while it is questionable whether a single matrix can effectively model complex spatial relationships. The authors need to provide further clarification or stronger evidence to support their claims.

2.The authors lack discussion of some existing work, for example, the paper [1] also focus on causal discovery in spatial-temporal data. It is essential for authors to further discuss the similarities and differences between the two papers and alleviate my concerns about the novelty of this work.

[1] Zhao, Yu, et al. "Generative Causal Interpretation Model for Spatio-Temporal Representation Learning." Proceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining. 2023.

1. The authors don't seem to mention how to guarantee that the learned graph G is a DAG, especially for the instantaneous relations.
2. The experiments are conducted on data with regular grids. How about irregular grids?
3. It may be more useful if the authors can further prove that the causal graph itself is identifiable.

The main weakness of the paper relies on identifiability. Below are all my concerns with regards to this matter. Other concerns will be written in the Questions part as I believe they are less important.

• Identifiability is introduced without a proper definition in the introduction (it appears in line 67 for the first time).
  • Identifiability should be defined along with corresponding citations to understand what is the equivalence you are looking for in your problem (I would assume graph structure).
  • It should also be accompanied with motivation on why identifiability is important, e.g. proving you have a unique solution up to some equivalence, scientific discovery (assuming assumptions in real setups hold), etc.
• Identifiability results are very weak considering the complexity of the data generating process. It only considers linear mixing which gives some affine identifiability (correct if mistaken), which feels redundant.
  • Line 315: "We focus on the specific case where no non-linearity maps the latents to the observable space". Then, Theorem 2 establishes identifiability up to " transformation by an invertible matrix". The result does not feel very strong as it does not seem to go beyond linear mixing.
  • Also given $Z\in\mathbb{R}^{D\times T}$, which is identifiable up to an invertible transformation, does this mean that theres mixing no only on $D$, but also over $T$? This should be clarified in the main text.
• Identifiability does not cover all the target latent variables:
  • Line 174: “Our goal is to infer the latent variables and the true causal graph in an unsupervised manner”. However, no identifiability results with respect to graph structure.
  • Generally I don't consider the graph structure to be identifiable considering identifiability up to a linear transformation (correct me if I am mistaken). Even with permutation and scaling identifiability, I don't see how instantaneous effects can be discovered with your assumptions.
  • For latent graph estimation:
    • In the absence of instantaneous effects, I would expect first identifiability of the latent variables up to permutation and component-wise nonlinearity.
    • With instantaneous effects, the task is even more challenging. Zhang et al 2024 recently has some results in non-temporal data.
• The paper generally does not acknowledge the above limitations, which I find an important concern given the attention it gives to the theoretical part.
  • Line 69: “our framework can handle both instantaneous edges and overlapping spatial factors”. Although it might be true empirically, this is not true with regards to identifiability. Therefore, I don't consider this claim to be valid.
  • Line 532: "We discussed the structural identifiability of our model". Structural identifiability as defined by Peters et al. 2017, refers to identifiability of the causal graph. Therefore, this is not true if the above points are correct.

Note: I would be happy to raise my score provided that this issue is addressed either by acknowledging this limitation in comparison to other works or establishing some stronger identifiability results (at least for the causal graph).

References

Zhang, K., Xie, S., Ng, I. & Zheng, Y.. (2024). Causal Representation Learning from Multiple Distributions: A General Setting. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:60057-60075 Available from https://proceedings.mlr.press/v235/zhang24br.html.

Peters, J., Janzing, D., & Schölkopf, B. (2017). Elements of causal inference: foundations and learning algorithms (p. 288). The MIT Press.