Paper:
We address the problem of reconstructing a matrix from a subset of its entries. Current methods, branded as geometric matrix completion, augment classical rank regularization techniques by incorporating geometric information into the solution. This information is usually provided as graphs encoding relations between rows/columns. In this work we propose a simple spectral approach for solving the matrix completion problem, via the framework of functional maps. We introduce the zoomout loss, a multiresolution spectral geometric loss inspired by recent advances in shape correspondence, whose minimization leads to state-of-the-art results on various recommender systems datasets. Surprisingly, for some datasets we were able to achieve comparable results even without incorporating geometric information. This puts into question both the quality of such information and current methods' ability to use it in a meaningful and efficient way.

Review:
This paper proposes a new method for geometric matrix completion based on functional maps. The proposed algorithm is a simple shallow and fully linear network. Experimental results demonstrate the effectiveness of the proposed method. 
The proposed method is new and has been shown good empirical results. The paper also points out a new way to interpret matrix completion. On the other hand, the proposed method seems ad hoc and there is no clear evidence why it is better than other baselines except the empirical results. The paper also has some clearance issues, making it hard to understand. I vote for a weak reject of the paper at the current pace and would like to increase my score if the following questions can be clearly answered.
1.	Why do we need to propose the algorithm? Is it because we have the functional maps technique motivated from shape correspondence, and we can see some connection of such technique with matric completion? If it is true, we surely can have a new algorithm based on such a new technique. But I can still not understand why the method work, at least, in an intuitive way.
2.	What is the sample complexity of the proposed matrix completion algorithm? 
The introduction of the paper is poorly written. The first paragraph and the third one both contain some introduction to matric completion, which results in a lot of redundant information. The second paragraph and the fourth one are redundant in the same way since they both focus on geometric matrix completion. I think besides introducing what is matrix completion and what is geometric completion, the introduction part should focus more on the motivation to propose the algorithm. However, I can only see from the end of the second paragraph (some simple models need to be proposed) and the fifth paragraph (“The inspiration of our paper”) some motivation information. The introduction part needs to be re-organized to provide more useful information about the paper rather than a literature review.
There is some unclear/inaccurate/subjective statement in the introduction part. For example, “Self-supervised learning” needs a reference. Why geometric matrix completion generalizes the standard deep learning approaches is not clear. What does it mean by “their design is … cumbersome and non-intuitive”? The shape correspondence is never explained until very later in the paper.  Also, there are some unclear issues besides the Introduction part. For example, what does it mean by “the product graph”? All these issues need to be clarified before the paper can be accepted. 
---------------------------------------------------
Thank you for the detailed rebuttal. For Q1, it clearly explains how does the method work. However, it is still not clear why does the method work. I also have another concern after reading the rebuttal, if the shape correspondence is not that important, why make it an important motivation in the paper? For Q2, it is interesting to see some theoretical results on the sample complexity, rather than an experimental one. The paper would also be much better if the clearance issues can be addressed. Even if I would not vote for an accept this time, I am looking forward to a revised version in the future.