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: This paper studies rank-1 matrix estimation and provides performance guarantees for two inference methods, a Bayes estimator and an AMP algorithm. 
strengths_and_weaknesses: 1. The author should comment on whether Assumption 3 holds for some representative examples like soft-thresholding function under the setting considered in this paper.
2. The results and quantities mentioned below Theorem 2 should be formalized into a theorem or corollary.
3. How crucial does Theorem 1 depend on the prior distribution on the signal X (line 163)? Given that the topic of this paper is to study whether these estimators are robust to wrong noise distributional assumption, it doesn't make sense if the result relies highly on the distributional assumption on the signal X (which seems to be an even stronger assumption that might not be correct).
4. When analyzing AMP, the authors assume access to an initialization that is independent of the noise and satisfy certain probabilistic condition (line 186). The authors also claim that in prior literature it is possible to analyze AMP with a practical spectral initialization (line 190). Given the assumed model is already simple enough (rank-1 spiked model), I recommend the authors to improve their analysis to adapt to such a practical initialization scheme.
5. I suggest the authors to make effort discussing their theoretical result in more detail and show how they are related to the main topic of the paper. Currently it is difficult to see this from Theorem 1 and 2.
6. If I am not missing something, the main result for spectral estimators is missing, although from Section 2.3 it seems that there should be some theory for spectral estimators.
questions: See "Strengths And Weaknesses".
limitations: See "Strengths And Weaknesses".
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: The paper considers the penalty to be paid when recovering a rank-1 matrix from noisy partial observations when the noise is assumed to be Gaussian in the case where the noise isn't so. The paper's analysis considers two algorithms and regimes: a Bayes estimator in an asymptotic regime and an approximate message passing algorithm in a state evolution regime. The analysis finds a gap in the performance of the two estimators that depends on the level of noise due to an inaccurate estimation of the signal norm in AMP.

strengths_and_weaknesses: The experimental results show a good match between the performance predictions of the analytical results and the true performance levels from the numerical simulations, and f the dependence of the performance gap between the algorithms on the SNR and noise model mismatch level.

Some instances of simplified notation do not seem necessary, and can detract from clarity (e.g., dropping the index N in line 171, the lack of description of the terminology used in Assumption 2). 

The discussion after Theorem 2 would be more clearly stated as a corollary: given that the statement of Theorem 2 appears to be very broad, the authors could provide a statement that is more relevant to assessing the performance of the estimators using metrics like those shown before.
questions: For the rescaled overlap measure (5), is it the same as the asymptotic the cosine distance between $\bf X$ and $\bf{\hat{x}}({\bf Y})$? If yes, this may be easier to describe. Additionally, the concern on the Bayes estimator in line 179 seems to be independent of the overlap measure - would it not also apply to the MSE?

Where is the function ht in (8) described? The sketch of the proof of Theorem 2 refers to "choosing denoisers carefully". Can there be a more explicit discussion of what "carefully" choosing denoisers entails in practice?

In line 254, should "converges... to the random vector" be "converges... to that of the random vector"? It may help with clarity to state at this point that the second random vector provide the state evolution for this AMP analysis.
limitations: The paper could use a more detailed discussion on the applicability of Assumptions 1, 2, and 3 in practical settings.
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: The authors considered the problem of rank-one signal estimation corrupted by structured rotationally invariant noise, and studied the situation when the actual noise statistics deviates from the Gaussian assumption. Three estimation procedures were proposed and discussed: mismatched Bayes estimator, approximate message passing, and spectral estimators. Both theoretical analysis and simulation experiments were provided to compare these different methods, together with reasonable conclusions and interesting insights.
strengths_and_weaknesses: Strengths:
1) A thorough study on the mismatched estimation problem for rank-one matrices was provided, with in-depth theoretical analysis and interesting empirical findings.
2) Different estimation methods were proposed and compared, showing the strengths and weaknesses of each method.
3) An interesting phenomenon was identified that Gaussian AMP does not perform as well as the mismatched Bayes estimator. Discussions were provided on potential causes of this phenomenon.

Weaknesses:
1) The Gausian AMP algorithm was provided in equation (8), but not much details were provided on how it is derived. It is unclear what objective function this procedure is optimizing. More technical details would help the readers to better understand the algorithm.
2) Similarly, the mismatch performance metrics were provided in equation (9), but not much details were provided on the meaning of these metrics. Better explanations of these metrics may help the readers to better understand the motivations behind the equations.
questions: The authors mentioned that the performance gap between the AMP algorithm and the Bayes estimator is due to an incorrect estimation of the signal norm. A natural question is whether the AMP algorithm could be modified accordingly, so that a better estimation of signal norm can be achieved, so as to narrow down the gap to the Bayes estimator.
limitations: The paper focused on theoretical analysis and simulated experiments. It would be nice if the authors can relate the research to some real world scenarios, and show some experimental results on real data.
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 presents a theoretical analysis of the performance of two algorithms for rank-1 matrix recovery under mismatched noise. Specifically, a closed-form MSE for the Bayes estimator and a state evolution analysis of the AMP algorithm are provided. The paper also discusses some phenomena arising from numerical experiments regarding these algorithms under two different performance metrics.
strengths_and_weaknesses: Strengths:
- This paper gives theoretically solid performance guarantees for the Bayes estimator and AMP algorithm for mismatched rank-1 matrix recovery. Settings with unknown noise distributions are common in real-world applications, where this work helps to understand the effect of these algorithms under mismatched noise. 
- Numerical experiments comparing the theoretical, experimental, and noise-aware performances are presented, which corroborate the results of this paper and shed light on the points of discussion.
- The writing and presentation of this paper is clear.

Weaknesses:
- Aside from the technical contributions, I am not fully convinced of the level of impact that this paper makes. This paper analyzes two existing algorithms with mismatched spherically symmetric noise, which is a step up from the analysis in [71] of mismatched Gaussian SNR, but might not suffice for this venue. Would consider raising the score if this concern is properly addressed.
questions: The discussion about Bayes estimator and AMP from the numerical results is interesting, but it might be helpful to provide more insights into the performance gap of AMP and Bayes in this problem setup, in addition to this phenomenon itself.

I understand that this is a theoretical work and if the authors think these are out of scope for this work, but I'd appreciate it if the authors can provide some comments about the applicability of the algorithms discussed in this paper. Under what type of scenarios might one favor one approach over the other?

limitations: While this is very theoretical work that proves performance guarantees, a dedicated discussion of the limitations of the scope and applicability of the results might still be helpful.
review_title: null

Meta-review: metareview: This paper studies precise high-dimensional asymptotics in a simple low-rank matrix estimation problem. When there exists a distributional mismatch between the true noise distribution and the Gaussian noise assumption imposed to run the AMP algorithm, the authors observe,  and formally quantify, the performance gap between the AMP algorithm and the Bayes estimator.  While the model considered in this paper might still be too idealistic (which limits its broader impacts), the paper is well-written and solid. All reviewers recommend acceptance of this paper, and I echo their recommendation.  
confidence: Certain
Decision: Accept