reviews: 
Comments: This work introduces a type of interventional extrapolation which consists in predicting the effect of an unseen intervention. The causal model considered is given by A -> Z -> X and Z->Y where Z is latent, A -> Z is linear, Z->X is invertible and Z -> Y is an additive noise model. The goal is then to estimate E[Y | do(A = a*)] where a* is outside its training domain. The authors separate this problem in three subproblems: 1) estimation given that Z is observed, 2) estimation this given that Z is known only up to an affine transformation, and 3) how to identify the invertible map Z->X up to an affine transformation. These identifiability results are then transferred into a practical algorithm: a) The mixing function is recovered by performing SGD on a loss consisting of a reconstruction term and a regularizer based on maximum moment restriction (MMR), and b) the learned encoder is then used to estimate the desired “do” expectation following the procedure of step 2) above.  The proposed estimation procedure is then validated on a synthetic dataset.

Review Summary: I gave the main text a thorough reading and checked the large majority of the math it presents. Despite the few problems I raised in my review, I believe this work is sufficiently novel, interesting, rigorous and well written to warrant acceptance to ICLR. I was planning to give a score of 7, since this score is not available, I'm rounding up to 8. I'm giving this score assuming that a discussion of the limitations will be added, as I suggest in my review.
Strengths: - I agree that we need more theory to demonstrate concrete benefits of identifiable representation learning. I enjoyed this perspective.
- The paper is very clearly written and structured and relatively easy to follow despite its technicality. The level of mathematical rigor in this work is very high in my opinion and the notation is always very crisp and transparent. 
- I thought the theoretical contributions were novel and interesting. 
- I appreciated that the main ideas about the proofs were provided in the main text. Most works in this literature only give very short proof sketches if any. 
- Theoretical results are well modularized to facilitate reuse.
- I thought the method to learn the encoder up to affine transformation was novel and interesting.
- I’m left with a good feeling of having learned something new.
Weaknesses: - The limitations of the proposed approach could be discussed further. For example, how does the method perform under various assumption violations? A few experiments could provide some insight into this. 
- The paper ends a bit abruptly. There’s no conclusion nor discussion of limitations. I guess this is the cost of having so much technical details in the main text (which, as I said, I appreciated, but I’m unsure whether this is a good balance.)
- I think the paper could do a better job of contrasting its results with what already appears in the literature, especially regarding the affine identifiability results. How does this compare to iVAE for example, which has a very similar data generating process with an auxiliary variable (which would be A here)? A few papers have similar results: [1,2,3,4].
- I couldn’t grasp a few reasoning steps in the main text (see Questions below).

Minor:
- $M_0$ has full row rank implies that dim(A) >=  dim(Z). Might be worth making explicit.
- I always forget which noise variable U or V is associated with which variable. How about replacing V by V_z and U by V_y?  
- Footnote one: what is $\mathcal{B}$?
- The introduction rightfully points out that the literature has almost no works giving theoretical arguments for why the identifiable representation learning is important. You might want to consider citing [1, 5] as examples of work pushing that direction.

[1] S. Lachapelle, T. Deleu, D. Mahajan, I. Mitliagkas, Y. Bengio, S. Lacoste-Julien, and Q. Bertrand. Synergies between disentanglement and sparsity: a multi-task learning perspective, 2022. 

[2] I. Khemakhem, D. Kingma, R. Monti, and A. Hyvarinen. Variational autoencoders and nonlinear ¨ ica: A unifying framework. In Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics, 2020.

[3] Ahuja, K., Mahajan, D., Syrgkanis, V., and Mitliagkas, I. Towards efficient representation identification in supervised learning. In First Conference on Causal Learning and Reasoning, 2022

[4] Roeder, G., Metz, L., and Kingma, D. P. On linear identifiability of learned representations. In Proceedings of the 38th International Conference on Machine Learning, 2021.

[5] S. Lachapelle, D. Mahajan, I. Mitliagkas, and S. Lacoste-Julien. Additive decoders for latent variables identification and cartesian-product extrapolation. In Advances in Neural Information Processing Systems, 2023b.
Recommendation: 8: accept, good paper
Reviewer confidence: 4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.
Soundness/correctness: 4 excellent
Clarity: 4 excellent