Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper presents a framework for unsupervised learning for inverse problems using multiple incomplete measurement models. The authors present theoretical results on number of measurements per model and number of models required for recovery assuming low-dimensional structure on the inputs. A training loss based on cycle consistency across different operators is suggested as a practical way to train these models. Experiments are performed to validate the theory and demonstrate on applications.
strengths_and_weaknesses: Strengths:
This paper builds off of prior work in low-dimensional signal recovery and very cleanly extends the results to fixed number of measurement operators and general low-dimensional structure. It also matches other known results like AmbientGAN for infinite measurement operators. In addition to developing a highly principled theory, the authors propose a novel and intutitive loss function that enforces a type of cycle consistency across operators. They show that this has a real effect on the result, compared to other schemes like SSDU that enforce cycle consistency across emeasurements. The paper is well written, the theory is well described, and the experimental results are strong. The 


Weakness:
There appears to be a disconnect between the proposed training method and the theory. In other words, no analysis, intutition, or experimental verification is provided on the multi-operator learning. The authors do not discuss or show any type of convergence results for the training method.

The authors appear to use different network architectures for AmbientGAN and MOI in the Inpainting CelebA section. This makes direct comparison difficult.


questions: Could you use the same network architecture for AmbientGAN as you use for MOI and supervised learning? This would aid the comparison.

It would also be instructive to experimentally see the impact of noise on recovery, as well as for different numbers of measurements per operator.

limitations: The authors describe the limitations of their work sufficiently.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper studies unsupervised learning from multiple incomplete measurements for linear inverse problems. It formulates an interesting and sound problem where observed samples $y_i$ come from multiple measurement operators but are not necessarily related to the same signal $x$. Such a problem setting is practical and widely occurred in many real-world applications such as medical, biological and scientific imaging tasks. This paper first established a theoretical framework to identify the necessary and sufficient conditions for unsupervised signal recovery, which could be viewed as a generalized results for blind compressed sensing. Then it introduces an unsupervised loss for learning from incomplete measurements, inspired from the presented theoretical results. The experiments on compressed sensing and inpainting of natural images as well as accelerated MRI consistently demonstrate the superiority of the proposed method against baselines. 

strengths_and_weaknesses: In my sense, this is a high-quality paper with both theoretical and empirical strengths. The studied problem setting is quite attractive and promising, which should be able to be applicable to a range of inverse problems in reality. The paper itself is nicely written and organized, even in the case fueled with mathematical details. I don't see severe weaknesses, though I do have questions/suggestion which might make the paper clearer and stronger [See Questions].
questions: [Theory part]  
The problem formulation assumes signals are observed via G linear operators, but I don't find any claims of characterization of this set of operators. Are there any restrictions on these G operators, e.g., linear independency?

[Experiment part]  
Could the authors provide a toy example in which the proposed unsupervised learning approach can compete with the supervised one, If the sufficient conditions in theory are meet. This would make the theory more convincing. 

[Related work]  
There are more related works studying unsupervised learning scheme for inverse problems [A,B,C]. Though the formulated problem settings are different, It would be very helpful to thoughly discuss the similarities and differences. Particularly, the loss function in [A] is very similar to the proposed one. Conducting ablation study with different unsupervised settings ([A] and this paper) and same reconstruction method (UNet) would be highly attractive. 

[A] Training Image Estimators without Image Ground-Truth, NeurIPS 2019  
[B] Unsupervised Learning with Stein’s Unbiased Risk Estimator, arxiv 2020  
[C] Training deep learning based image denoisers from undersampled measurements without ground truth and without image prior, CVPR 2019  

Overall it's a promising work from my perspective, and I would raise my score if the above issues could be addressed sufficiently.
***
**The rebuttal addresses my concerns, thus I raise my score to Accept**
limitations: N/A
review_title: null

final_score: rating: 3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.
review_content: summary: This submission claims to characterize a sufficient condition ("model" being low dimensional) for the linear signal reconstruction problem to have the unique solution (?) in the unsupervised learning setting.
strengths_and_weaknesses: - The narrative is unclear as there is no clear statement on the setup of the main problem under consideration.
- It is likely that the authors mistake the term "model" with "signal".
- The statement of main results is vague.
- Numerical evidence is unconvincing.
questions: - What is the setting of the unsupervised learning problem? There is no specification given in the submitted manuscript that clarifies this.

- Can authors elaborate what 'low-dimensional models' mean in this manuscript? This is unclear in theorem 4.2.

- In numerical experiment, line 288, what are the theoretical bounds authors are referring to?
limitations: N/A
review_title: null

Meta-review: metareview: This paper proposes an unsupervised learning algorithm for inverse problems using multiple incomplete measurement models. The authors presented theoretical results on the number of measurements per model and the number of models required for recovery under the assumption that the inputs have low-dimensional structures. In addition, a conceptually simple unsupervised learning loss is proposed and it only requires access to incomplete measurement data. The paper is well written, its theoretical analysis is strong, and the experimental results are convincing.
confidence: Certain
Decision: Accept