Venue: NeurIPS 2022 Conference

Reviews:

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: The paper presents an approach to improve CKA and RSA, which are the representational similarity metrics. Such metrics are used to compare representations from different layers of the same/different networks given the same list of objects.
Authors propose to "regress out"  inter-object similarity, the idea came from biostatistics.
Experiments with comparing networks trained from different seeds, fine-tuned networks, transfer learning show that the "deconfounded" CKA  (dCKA) is more sensitive than regular CKA and better aligns with the intuitive notion of functional similarity.
The paper is well written and easy to follow.
strengths_and_weaknesses: Strengths:
* interesting idea regarding deconfounding.
* convincing experimental results from vision and text domains

Weakness:
* I wish authors explain the intuition behind deconfounding in more details. 
* some experimental details are not presented - how do you choose kernels in dCKA?
questions: 1. eq. (4) what does vec(*) mean? I suppose it is matrix flattening? Maybe it makes sense to explain it explicitly.
2. The paper "Similarity of Neural Network Representations Revisited" describes an interesting experiment, 
where the goal is to find the most similar layer between two networks trained from different seeds.
Ideally, the most similar layer should have the same number. 
It is very interesting to evaluate your dCKA metrics for this problem.

3. line 132: "we can approximate dK_{f_1}^{m_1} and  dK_{f_1}^{m_1}" - you repeated dK_{f_1}^{m_1} two times.
4. in eq (3) you use an additional kernel to "regress out" the input similarity.  It should bw different from the original one in CKA. 
Which kernels you used in the experiments?
limitations: No potential negative societal impact from this paper, in my opinion.
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: The paper investigates the confounding effect of similarity metrics used to compare neural networks on a layer-by-layer basis. The authors study the effect of this confounding effect. They argue that such measures as CKA and RSA are affected, thus, miscommunicating the functional similarity information by showing that the measures can show high similarity even for random neural networks. The paper proposes to 'deconfound' the metrics using a simple fix by regressing the input similarity structure from the representation similarity structure. Next, the authors conduct extensive experimentation to show the deconfounded measure performance under various settings. 
strengths_and_weaknesses: Originality
The paper tackles an interesting problem, though, relatively narrow in scope, but still valuable for a meta or transfer learning domain. While the correction of the similarity metrics is not improving on models' performance, it can provide important insights into the inner workings of these models.   

Quality
The paper is well-balanced and has all the necessary components: it introduces the problem and supports it with experiments, it solves the problem and provides necessary proofs without overburdening the paper. A multitude of experiments is introduced showing various aspects of the problem. The code and appendix are also provided. While some experiments are not straightforward and nested (having multiple steps in them on top of each other), I appreciate the complexity of the validation of the similarity metrics and how 'functional' similarity is difficult to define. 

Clarity
It is a very well-written paper that is easy to read. It does not overcomplicate the idea, and the solution is quite elegant. I liked the experiments that provide interesting insights into neural networks' inner workings. 

Significance
A relatively simple fix of the similarity metrics correction is not necessarily a major contribution that will lead to a groundbreaking impact in ML area, but it is a solid paper that helps to address a problem for future work on models' interpretation and lead to the better model training experience. 
questions: I don't have many questions as mostly the paper is well-written and provides all the necessary information. Just a few comments:

- In 3.3 you do not mention/link to the proofs of the proposition provided in the Appendix
- In figure 2 x-axis label is missing. Also, the colouring of Random NN-s is a bit difficult to see
- In 4.1 you perform a lot of experiments that include hypothesis testing and statistical significance. Have you considered multiple testing corrections?
- In Figure 5 (B) the colours for zoom blur and contrast are mislabeled. 
limitations: No limitations detected
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 aims at measuring the _functional_ similarity between two given neural networks.
The existing approaches naively compute some similarity measure of intermediate outputs of two neural networks, which may be largely entangled with input data, leading to spurious similarity.
The authors propose a method to deconfound the effect of input data by simply regressing out their effect.
The decounfounded similarity measures admit a good correlation with domain similarity and OOD accuracy, which are observed through experiments.
strengths_and_weaknesses: ### Strengths

1. The spurious correlation between input data and functional similarity is formulated as a linear regression model. Although this simplicity may discard several more intricate structures, the deconfounded solution can be obtained by simply solving the normal equation. This formulation would be sufficient for the first step towards the deconfoundation.
2. The proposed method basically can be applied to any type of the existing representational similarity measures by applying the deconfoundation step. Through the experiments, we can observe that adding this simple fix is beneficial to improving the performance in most cases.

### Weaknesses

1. The motivation to measure the representational/functional similarity could be made a little bit clearer. For example, Section 4.3 provide the experiments to observe the correlation between the functional similarity and domain similarity, and Section 4.4 to see the correlation between the functional similarity and OOD accuracy. While these results may seem excellent, we would not immediately see what such a functional similarity brings us to improve transfer learning and OOD generalization.
2. The functional similarity is basically measured between two layers of given neural networks, not the entire neural networks. To measure the similarity between the entire networks, the authors simply average the layer-wise similarity through the experiments. Whereas this works in some situations (for example, Figure 2 tells us the averaged similarity is sufficient to distinguish random and fine-tuned ResNet-18), I have a concern when networks are deeper; most of the high-level features are similar between two networks, and hence the layer-wise averaged similarity would not be sufficient.
questions: ### Questions

1. In Section 2.1 (prior work), the two-staged similarity measurement method is introduced (Eqs. (1) and (2)), but it may be not that evident why the first stage (Eq. (1)) is needed. A few more explanations would be preferred.
2. I am not sure why the invariance properties discussed in Section 3.3 are important. In the propositions, the proposed similarity measures are shown to be invariant against orthogonal transformations and isotropic scaling, which should not change the inter-example similarity. In my understanding, a good functional similarity should be independent of the inter-example similarity. If the aforementioned operations do not affect the inter-example similarity, the importance of the invariance would be questionable. Hence, I would like to see some clarification.

### Suggestions

1. In Eq. (2), do we need the full generality to introduce the similarity between different layers such as $m\_1$ and $m\_2$?
2. (minor) At l.88, "decounfounded" -> "deconfounded"
3. (minor) In Section 3.1, you may define $K^0$ properly right after Eq. (3).
4. (minor) In Figure 4 (right), having the averaged similarity in the legends (like Figures 2 and 3) should be better.
limitations: The authors adequately addressed the limitations in the discussion. The linearity assumption is one of the biggest limitations of this work, but the proposed method still works fairly in several experiments. This work may not be relevant to any negative societal impact.
review_title: null

Meta-review: metareview: The paper makes the observation that neural network similarity indexes can be misleading when compared across domains with different examples. The paper presents a fix via covariate adjustment, which improves quality of similarity indexes across neural networks across domains. The approach is simple, and the reviewers unanimously agree that the paper is worthy of publication at NeurIPS.
confidence: Certain
Decision: Accept