Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This paper proposes a relatively novel method using shape metrics to measure similarity across representations in both neural networks and in the brain, which is an important problem in both fields. The paper compares its method to existing representational metrics and shows that it is able to overcome some of the previous limitations as well as to provide new insights from representations. 
main_review: Originality: The incorporation of shape metric into analyzing representation similarity is novel to my knowledge. 

Quality: The proposed shape metrics is well motivated and mathematically sound. Given that the authors shows that their method could give different results in hierarchical clustering of representations, which by itself is very interesting, it is unclear how differently it does in other applications comparing to existing methods. The persistent structure across layers, as well as the relationship from test set accuracy and representations might be visible using existing representation similar metrics. I would like to see more experiments done to further validate the method. For example, it can be used to compare representations learned for different tasks to inform multi-task learning, or on a different brain dataset. 

Clarity: The paper is clear and well written. 

Significance: Application of the method proposed by this paper is very general and extremely useful to both neuroscience and machine learning. 
limitations_and_societal_impact: Yes.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The current manuscript presented a unifying framework to analyze the representation similarity between neural networks. The authors first established that the distance metrics between neural representations should satisfy equivalence, symmetry and triangle inequality. And the authors showed that a simply modification of canonical correlation analysis would satisfy these reequipment. Next, the authors identified the transformation matrix that best conform the two representation spaces under various conditions. The current framework revealed the hierarchy of multiple brain areas and visualized the representation structure of a deep neural network.  
main_review: In the current manuscript, the authors provided theoretical foundation for representation similarity analysis, as well as demonstrated its application in both real and artificial neural networks. While the methods seem to be not novel, the theory was valuable and the application results were intuitive. However, it was confusing that mathematical foundations did not lay out in a way to guide the applications. It seems extra math were provided without giving an example (eg. the permutation invariance, rotation invariance and so on), while the real-world application seems to only relevant to a small fraction of the theory.   
limitations_and_societal_impact: The authors discussed that limitation of representation analysis and acknowledged that the representation similarity was only loosely related to network function.
review_title: null

final_score: rating: 9: Top 15% of accepted NeurIPS papers, strong accept
review_content: summary: The paper is concerned with issues that are fundamental to basic research on understanding natural and artificial neural networks. Such research often involves measuring the similarity of neural representations (patterns of neural activity). The authors propose desirable attributes for the methods used for such comparisons, e.g. that the similarity function should define a metric metric space: it should satisfy the constraints of equivariance, symmetry and the triangle inequality. They show that many commonly used methods do not satisfy these constraints but can be easily modified to do so. They also construct a metric especially suited to convolutional layers, which traditionally have not been straightforwardly analyzed by existing methods. The authors also discuss practical issues of general importance: They discuss what sample size is required for such metrics to be accurate/robust, which is highly important for neuroscience research where it can be difficult to obtain large sample sizes. They also talk about dimensionality reduction, showing that approximating neural embeddings with low dimensional Euclidean spaces, while suitable for visualization, can highly distort the embedding. They validate their new metrics by showing that it can recover known structure in large scale datasets.
main_review: The work is important, timely, clearly written, and likely to be impactful in both neuroscience and machine learning. It addresses an important issue that I have previously noticed in the work on similarity of neural representations. The analysis appears to be appropriate and correct. The work is clearly motivated and the results actionable. I'm afraid I have nothing constructive to add. My only notes on the paper are positive. 
limitations_and_societal_impact: The authors have identified important limitations of their work and highlighted directions for future work. They do not address potential negative societal impact but I can offer no suggestion in that regard. The work does not appear to have any obvious potential negative societal impacts.
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This paper develops metrics for comparing different (yet presumably related) representations. The emphasis is on representations generated by artificial or biological neural networks. The paper relates this development to previously proposed approaches. Experimental results on recordings of neuronal activity in the mouse brain and representations generated by different layers of different deep neural networks are presented.
main_review: - It is not clear why representations of different networks should be linearly related. It seems identifying topological similarity would be a better fit for this problem, although it may be computationally impractical. I find the reasoning behind the equivalence classes being defined through linear transformations missing. At least, a topology-centered introduction which discusses its practicality before settling on linear transformations would be a better exposition.
- It takes a while to realize that this work concerns the alignment of representations and these representations do not have to come from networks. Network representations is the application here, although the development does not have much to do with having an underlying biological/artificial neural network.
- The paper devotes ~5 pages to reviewing the basics and setting up the problem. As a result, the experimental section is somewhat rushed and whether the approach offers practical improvements is not clear. Although, of course, having a principled approach is inherently advantageous.
-  On a related note, I couldn’t tell if the authors claim novelty about the propositions. They are rather elementary, including the generalized version in the supplement, which covers both cases. It wouldn’t surprise me if they appear in a textbook etc although I can’t cite one here. Please clarify this point.
- The experiments would be much more meaningful in the presence of baselines. For instance, if the ABO experiment were repeated, except with non-metric embeddings, but following the same multidimensional scaling and PCA parameters, how different would the embedding of the brain regions be?
- Why does it make a big difference to embed in L-dim space first and then perform dim. reduction rather than embedding immediately?

Minor:
- (line 120) Please define the equivalence relation.
- (line 31) As written, the definition corresponds to a semi-metric. Please include the equivalence condition here as well.

Clarity: The exposition is generally clear and it is not hard to follow the development.

------------------------------ UPDATE ------------------------------

I decided to increase my score from 5 to 6. Please refer to the discussion with the authors for details on the topological aspects of the problem. 
limitations_and_societal_impact: No, but I don't think this mathematical development carries foreseeable potential negative societal impact.
review_title: null

Meta-review: This paper introduces a new similarity measure for representations. Given the wide recent use of comparing representations both in analyzing neural networks and in analyzing brain activity. There was enthusiasm from the reviewers about the fact that the metrics introduced are formally based, but however, there was a discussion about whether the paper should include a more formal topological analysis. 

A consensus was reached by having the authors discuss their specific assumptions, the potential connections to topological analysis and what could be addressed in future work. Given that the authors have promised to implement these changes, I recommend acceptance. 

Decision: Accept (Poster)