Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: Two common ways of avoiding the non-identifiability in nonlinear ICA are 1) conditioning on auxiliary variables, and 2) staying in the unconditional setting but instead restricting the function classes.  This paper focuses on the latter (i.e. unconditional ICA), and studies the identifiability for conformal maps, orthogonal coordinate transforms (OCTs), and measure-preserving maps.

Specifically, the results are (summarized in Table 1):
- For conformal maps, the paper extends Hyvarinen & Pajunen 1999's results from d=2 to d>2.
- For OCTs, the paper proves a weaker notion of "local identifiability", and conjectures that ICA would be identifiable for "typical" pairs of elements, though the formal statement is left for future work. In addition, local identifiability does not hold for nonlinear maps in general, highlighting the importance of constraining to OCTs.
- Measure-preserving maps are not locally identifiable and hence also not identifiable.

Proof technique wise, the paper draws connection to rigidity theory and tools from PDEs.

strengths_and_weaknesses: Strengths
- The paper provides an alternative lens (i.e. restricting the function class) to circumvent the non-identifiability problem in ICA.
- The paper draws connection to techniques in PDEs. Concretely:
  - The local identifiability result in Theorem 4 follows from first casting the independence conditions of ICA as second order PDEs and then applying the uniqueness of the Cauchy problem of second-order hyperbolic PDEs.
  - The constraints in Theorem 5 are because the Dirichlet problem may not be well posed in general and doesn't always have a unique solution, unless under some specific constraints.
- The paper is clear about technical limitations, provides partial results whenever possible, and provides interesting open directions.

Weaknesses: I have no major concerns about the paper. I should note that I'm not the best judge for the impact of the results and the technique novelty.

questions: - Thm 3: Noting that $\mathcal{S}_{\text{conf}}$ contains a shift, could you explain how is this stronger than the identifiability result in Hyvarinen & Pajunen 1999?
- Clarification: Thm 8: does $|\cdot|$ denote the l2 norm?

=== Update after discussion ===
The authors have answered this in the reply.
limitations: The paper discusses the limitations well.

There's no direct societal impact.
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: This paper studies the identifiability of nonlinear ICA by focusing on different types of function classes (i.e., conformal, orthogonal, and volume preserving maps). In particular, the paper extends previous results of dimension 2 to show that ICA with conformal maps is identifiable for dimension larger than 2. The authors then define the notion of local identifiability and show that ICA with orthogonal maps is locally identifiable. Finally, the authors show that ICA with volume preserving maps is not identifiable. Throughout the theoretical analysis, the authors draw connection to the rigidity theory.
strengths_and_weaknesses: Strengths:
- The notion of local identifiability is novel to the best of my knowledge
- Theoretical analysis for different function classes and the connection with rigidity theory are provided and appear to be sound

Weaknesses: Overall, while this paper provides theoretical analysis for nonlinear ICA, in my opinion it will be helpful if there are some empirical studies to illustrate the theoretical claims. Details comments are as follows.
- Since the concept of local identifiability is new, it would be more appealing to provide more explanations regarding the intuition of local identifiability (e.g., how it specifically contributes to one identifying the desired independent components in practice), and empirical studies to demonstrate its practical applications. This may be necessary for others to better understand the significance of local identifiability. On the other hand, for the empirical studies, how does one evaluate the performance of an algorithm w.r.t. local identifiability?
- Related to my point above, more explanations (e.g., in what situation one will expect the corresponding function assumption to likely hold) and empirical studies for the function classes will also make the contributions more convincing.
- While the theoretical analysis of identifiability for conformal map might be potentially interesting, it might be rather restrictive and less likely to hold, and thus might limit its practical applications.

-----
After reading the responses and discussion, including the further comments from the authors and also the other reviewers, my concern about the applicability of the assumption persists.
questions: As noted in the "weaknesses" section above, it is helpful to provide more explanations and empirical studies regarding local identifiability and the different function classes.
limitations: Yes
review_title: null

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: This manuscript extends the previous identifiability result on conformal maps where the dimension is fixed to two. Specifically, The main result is that the Mobius transformation, which is a conformal map where the dimension is larger than two, is identifiable up to trivial indeterminacies. Besides, partial/local results on the orthogonal coordinate transformation (OCT) and unidentifiability of volume-preserving transformation have been presented. No experiments have been conducted.

The theoretical contributions could be summarized as follows:

1. (Main result) Identifiability of high-dimensional conformal maps.
2. Unidentifiability of OCTs and volume-preserving transformation.
3. New notion of local identifiability and its connection with OCTs.
strengths_and_weaknesses: Pros:

1. The paper is well-written.
2. Theoretical results are clearly and rigorously presented. The proof seems rigorous and its technique provides novel insight.
3. The non-identifiability of OCTs and volume-preserving transformations are helpful and consistent with intuition.
4. Limitations have been clearly and insightfully discussed.
5. The notion of local identifiability is original and interesting.

Cons:

1. The core assumption of the main result, i.e., high-dimensional conformal map, seems to be impossible in practice. In [1], OCTs (orthogonal columns of the Jacobian of the function) have been shown to be helpful to the empirical identification of nonlinear ICA. Although no identifiability result is given in [1], the corresponding intuition of independent mechanism could be found in the real world. However, compared to OCTs, high-dimensional conformal map restrict the Jacobian of the mixing function to be a scaled version of an orthogonal matrix, where the L2 norms of the columns and rows are exactly identical to each other, which seems to be impossible in practice.

2. Lack of discussion of the assumption, especially for its motivation and rationality. In L153, the motivation of OCTs has been referred to [1]. However, as described above, there exist key differences between high-dimensional conformal maps and OCTs, where the former seems to be impossible in practice. Therefore, it may be better to discuss the assumption of the main results, perhaps from a perspective of the chance/scenario of it being true. I couldn't find a related discussion in the current manuscript.

3. Lack of experiments. According to the reasons mentioned above, it is hard for me to check the significance of the main result. It may be of independent theoretical interest to study the identifiability of a specific functional form, i.e., the high-dimensional conformal map (or Mobius transformation). In that case, simulations focusing on the validity/contribution of the proposed assumption should be very helpful to support the claims. In the current manuscripts, no experimental results are presented.

4. Although OCTs (or IMA [1]) and volume-preserving transformation [2] have been proved unidentifiable in the manuscripts, there exists previous work that provides the full identifiability result with assumptions only on the mixing function. For example, [3] prove that, under assumptions of independent influences and factorial changes of volume, unconditional ICA could be identified up to trivial indeterminacies as the same as those shown in this manuscript. The assumption of independent influences seems to be closely related to OCTs and the assumption of factorial changes in volume contains volume-preserving transformation as a special case.  Therefore, it seems that the combination of the principles of the two assumptions discussed in this paper is sufficient for the identifiability of unconditional ICA. Thus, I believe the identifiability result of unconditional ICA in [3] is related to the task of this manuscript, which is currently missed in the discussion of related works.


-------

[1] Luigi Gresele, Julius von Kügelgen, Vincent Stimper, Bernhard Schölkopf, and Michel Besserve. Independent mechanism analysis, a new concept?

[2] Xiaojiang Yang, Yi Wang, Jiacheng Sun, Xing Zhang, Shifeng Zhang, Zhenguo Li, and
Junchi Yan. Nonlinear ICA using volume-preserving transformations.

[3] Yujia Zheng, Ignavier Ng, and Kun Zhang. On the identifiability of nonlinear ICA with unconditional priors.
questions: Please refer to the above section for the corresponding questions. I will change the score if the related confusion can be addressed during the rebuttal.
limitations: The authors discussed the limitations of the work.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper gives some deep insights for the identifiability of the limited function classes between linear non-Gaussian ICA and unrestricted nonlinear ICA, including conformal maps, orthogonal maps and volume preserving maps.
strengths_and_weaknesses: Overall, the paper is well written and easy to follow. I am no expert in this area, but as far as I can tell, the theoretical contributions appear sound, the quality of writing is generally strong. I believe it work will be a welcome contribution to the identifiability of nonlinear ICA. I am just confused by the real application scenarios of these function classes mentioned in this paper. It would be great if the authors can give a discussion about the real application scenarios.
questions: See above
limitations: NA
review_title: null

Meta-review: metareview: The paper studies identifiability of ICA for two families of non-linear functions: conformal maps and orthogonal coordinate transforms. For conformal maps, they prove identifiability for d > 2, improving an old '99 result for d=2 due to Hyvarinen and Pajunen. For orthogonal coord. transforms, they prove a weaker notion of "local" identifiability. 

There was quite a lot of discussion on the various strengths and weakness of the paper. 

(1) *Experiments*: Though the paper had very little experiments, the reviewers agreed that since the paper is primarily a theory paper, an extensive experimental section is not necessary.

(2) *Theory*: The reviewers found both the proofs of Theorems 2 and 3 quite interesting, involving new ideas. They're heavy on tools from complex analysis, which is not surprising giving that conformal maps are natural through the lens of complex analysis; but they found the connections to PDEs interesting and potentially useful in the future. There were some potential worries about correctness, but no definite error was identified. 

(3) *How strong of an assumption is conformality in practice*: The reviewers agree this is probably quite restrictive as an assumption, but idenfiability of non-linear ICA is always going to require some strong conditions, and we're still very far from understanding when it's possible (when no auxiliary variables are involved). The paper shrinks the gap b/w theory and practice even if the theory has very strong assumptions.
confidence: Certain
Decision: Accept