We thank the reviewer for their encouraging feedback, and we are pleased that they found our results relevant and meaningful.

Clarification on the assumptions of our theorems:

A central assumption of our work is the diversity condition (Definition 2.1). As the reviewer rightfully noted, this condition is more likely to hold for language models, whereas for regular image classification models, it might not be satisfied (especially when the number of classes is smaller than the representation dimensionality). We will clarify this point in the paper. As we noted in Section 6, recent work [1] may serve as a starting point for exploring how to proceed when the diversity condition does not hold, potentially extending the results to typical image classifiers. However, we leave this direction for future work. We remark that Theorem 4.6 has an assumption slightly stronger than the usual diversity condition (all $\mathbf{L}, \mathbf{L}’, \mathbf{N}, \mathbf{N}’$ constructed from the input and label sets are invertible). However, we expect that if the diversity condition can be relaxed, then this slightly stronger assumption may also be relaxed while retaining a qualitatively similar result.

“The bound on dSVD [...] is not very tight”

We agree with the reviewer’s observation and remark that this point is also acknowledged in Section 6 (“Limitations” paragraph). Nonetheless, we believe this result provides an important proof of principle: It demonstrates that representational dissimilarity can be bounded using a distance between distributions---a fact that is far from trivial, as Theorem 3.1 illustrates. Our experiments on synthetic data indicate that this bound could potentially be tightened; however, determining how to achieve this is left for future work.

“Validation in higher dimensional cases or on a larger dataset (i.e., ImageNet) would be a valuable addition”

We thank the reviewer for the positive feedback on our experiments: “While not large scale, they are well-designed to illustrate the argument.” We agree that expanding to larger-scale experiments is an important step for future work, especially to bridge the gap to practical applications. Furthermore, we are particularly interested in experimenting with language models, as they appear to align most naturally with our assumptions.

“accelerating / approximating LLV computation”

We hope that in future work we can reduce the number of labels required so that the metric scales with representational dimension rather than the number of labels. This would significantly reduce computation time for language models where the number of tokens far exceeds the representational dimension (e.g., GPT-2 has ~50k tokens vs. ~1000 dimensions). Approximating the computation of $d_\mathrm{LLV}$ is another promising direction for future work and could make it more suitable as a loss function.

Considering other f-divergences:

One of the reasons why the KL divergence cannot be used to bound representational similarity stems from the fact that unlikely labels (with $p(y|\mathbf{x})$ very small) give a small contribution to the KL, which enables constructions as the one in Theorem 3.1. We expect that other divergences with similar behavior would also not allow us to derive bounds on representational similarity. Conversely, divergences that do not put particularly low weight on unlikely labels (like $d_\mathrm{LLV}$) might be better suited for deriving bounds on representational dissimilarity, as in our Theorem 4.6. We plan to test other divergences in future work.

“Any thoughts on whether LLV could be used as an explicit penalty (loss) in something like distillation?”

We thank the reviewer for sharing this thought. We agree that this is an interesting direction to pursue, and we explicitly mentioned distillation in the introduction. However, we believe further research is needed before recommending log-likelihood variance distance as a loss function. We hope it will be possible to either find a simplified version of the distance or identify an upper bound or approximation that can serve as an effective and efficient distillation loss.

[1] Marconato, E., Lachapelle, S., Weichwald, S., & Gresele, L. All or none: Identifiable linear properties of next-token predictors in language modeling. AISTATS 2025

We thank the reviewer for their encouraging feedback and for acknowledging the strength of our theoretical results.

“What would we miss by just using CCA?"

This is an important technical point. To summarize:

(1) Similarity measures based on CCA take only pairs of embeddings, or unembeddings, as input, whereas our $d_{\mathrm{SVD}}\left(\mathbf{L}^{\top} \mathbf{f}(\mathbf{x}), \mathbf{L}^{\prime \top} \mathbf{f}^{\prime}(\mathbf{x})\right)$ takes both the embeddings and the unembeddings of both models as inputs, since the matrices $\mathbf{L}$ and $\mathbf{L}^{\prime}$ depend on the unembeddings $\mathbf{g}$. Note in fact that the distribution of the model (Equation (1) in the paper) depends on both the embeddings and the unembeddings.

(2) CCA-measures have thus different invariances and can lead to different conclusions than our dissimilarity measure $d_{\mathrm{SVD}}\left(\mathbf{L}^{\top} \mathbf{f}(\mathbf{x}), \mathbf{L}^{\prime \top} \mathbf{f}^{\prime}(\mathbf{x})\right)$. For example, mCCA between model embeddings is invariant to any linear transformation of the embeddings. In principle, one could find two models giving rise to different distributions where the embeddings of a model are a linear transformation of the embeddings of a second model, whereas their unembeddings are not equal up to the same linear transformation. These two models would thus have maximum mCCA (equal to 1) between embeddings, but they would have a positive $d_{\mathrm{SVD}}\left(\mathbf{L}^{\top} \mathbf{f}(\mathbf{x}), \mathbf{L}^{\prime \top} \mathbf{f}^{\prime}(\mathbf{x})\right)$ (not the minimum value of zero).

Thanks for prompting this explanation: We will clarify this in the revised version of the paper, also based on a discussion of (2) we currently have in Appendix E.8.

“...does the pivot… influence the distance or log-likelihood variance distance terms?”

We thank the reviewer for pointing out that this requires clarification. When models have different distributions, the choice of pivot (as well as the input and label sets) can indeed affect the value of the distance $d_\mathrm{LLV}$. As long as the diversity condition is satisfied, any choice of pivot is valid. If, on the other hand, the distributions of two models are equal, then the distance $d_\mathrm{LLV}$ will be zero no matter the choice of pivot and sets. We will make this clearer in the paper. In the experiments, we draw a sample of possible inputs with pivot sets and choose the set that gives the smallest value for the term $t_2$ in Definition 4.4 (we specified this in Appendix E.3: Implementation of the Log-Likelihood Variance Distance). We have not yet analyzed the effect of the pivot point probability on the value of the $t_2$ term in $d_\mathrm{LLV}$. It might be interesting to explore this in future work.

"Shape metrics from Williams et al…”

We thank the reviewer for this interesting reference and will include it as related work. We will also consider whether its ideas can inform the design of other similarity measures with the desired invariance properties.

We thank the reviewer for their constructive feedback and for recognizing the potential of our analysis.

“all embedding spaces are restricted to 2D”

Regarding the CIFAR-10 experiments in Section 5.1: to guarantee the diversity condition, the representation dimension needs to be smaller than the number of labels, and we specifically chose a dimension of 2 for ease of visualization. This makes it easier to compare our findings in Figure 3 (Left) with Figure 2, which illustrates the construction used in Theorem 3.1. To address the reviewer’s concern, we will add results from models we have trained on CIFAR-10 with 3- and 5-dimensional representation spaces. For 3- and 5-dimensional representations, we cannot provide a visualization like in Figure 3 (Left). Instead, we will include scatterplots in the appendix showing the difference in test loss versus the mean canonical correlation (mCCA) of embeddings from models trained on CIFAR-10. In these scatterplots, we see that it is possible to get both smaller loss differences and less similar representations, and models with higher loss difference and more similar representations. For 2 and 3 dimensions, we see that the minimum representational similarity is lower than for 5 dimensions, and the maximum similarity achieved is slightly higher. All in all, these scatter plots suggest that a small loss difference does not imply that the underlying representations are similar, in agreement with our Corollary 3.2.

“The experimental scope is limited”

The primary goal of our work is to deepen the mathematical understanding of representational similarity through the lens of identifiability. We appreciate reviewer r8r8’s comment that “while not large scale, [our experiments] are well-designed to illustrate [our] argument,” and we believe that the additional results described above and prompted by this review further strengthen our experimental validation. We agree that an empirical study with larger-scale experiments and models is an important direction for future research.

“CIFAR point clouds… the [embedding] norms might be fixed for these experiments”

We thank the reviewer for pointing out that this aspect of the code was unclear. We did not constrain the embedding or unembedding norms of the models presented in the main paper: The embeddings and unembeddings are plotted in Figure 5 to Figure 14 in Appendix F, and it can be seen that their norms are not fixed. However, we have some additional experiments in Appendix F where either embedding or unembedding norms, or both, were fixed. These ablations were not included in the main paper, but only in Appendix F.5, "Illustration of Bound on Constructed and Trained Models". We will clarify this point once we publish the code.

I might be misunderstanding: it seems that the authors are suggesting the visualizations in Figs 3 and 5-14 (where the image embeddings clearly have different radii) are evidence that the vector norms were not fixed when calculating likelihoods. The likelihood calculation in the code is not simply the inner product between the embeddings and the coembeddings: it has optional normalization in model_class.py that was turned on for some of the experiments specified in ARTICLE_model_variations_config_resnetcifar10_128_fd2.json.

To the contrary, the density of points near the origin and the crisp azimuthal decision boundaries suggests the opposite, that cosine similarity was effectively used instead of the inner product described in the paper.

def dot(self, features, target):

if self.fix_length_gs > 0:

current_fix_f_option = model_var_config.fix_length_gs[0]
current_fix_g_option = model_var_config.fix_length_fs[0]

We thank the reviewer for their helpful questions and for emphasizing the significance of our research question.

“Construction-centric nature of Theorem 3.1…”

It is true that just because these constructions exist doesn't mean a model will learn them during training. We will make this clearer in the article. However, identifying these examples is essential for mathematical understanding of the problem: identifiability research has similarly made progress by starting from constructed counterexamples (e.g. [1]) to understand non-identifiability issues. Moreover, we find it interesting that a mechanism resembling our construction actually emerged in the CIFAR-10 experiments (Section 5.1). Understanding when and why such constructions are learned in practice would be an interesting direction for future research.

“Permutation based representation dissimilarity…”

We appreciate the comment and suspect the confusion may be due to unclear phrasing on our part. The permutation in 3.1 should be understood as a permutation, $\pi$, of the label indices, $\mathbf{g}’(y_i) = \mathbf{g}(y_{\pi (i)})$ with a corresponding shift of the embedding clusters; and not as a permutation of the representation dimensions (i.e., the axes in the figure), $\mathbf{f}’(\mathbf{x}) = \mathbf{P}\mathbf{f}(\mathbf{x})$ for some permutation matrix $\mathbf{P}$, which would indeed be a linear transformation, as the reviewer points out. Unlike the latter, the permutation of label indices, and corresponding shift of embedding clusters, is not a linear transformation of the representations: In Table 1, we show that when constructing models in this way, the mean canonical correlation is close to 0, which means their representations are very far from being linear transformations of each other. The permutation of label indices also changes the model’s output distribution, since it alters the rank—by predicted probability—of every label except the top one for each input $\mathbf{x}$. Conversely, permutations of embedding neurons $\mathbf{f}’(\mathbf{x}) = \mathbf{P}\mathbf{f}(\mathbf{x})$ (and accordingly of unembedding ones) in a way that lead to the same probabilities will not alter the model’s output distribution and will thus yield equivalent model instances according to our identifiability perspective (models that have equal distributions are deemed equivalent). We will include the clarification above in the revised version of the paper.

“The authors argue based upon their defined measure that representational similarity increases with width, an arguably known result from past work”

We believe this comment may reflect a misunderstanding of the primary novelty of the experimental findings in Section 5.2. That higher-capacity models tend to produce more similar representations is indeed a known result, which we acknowledge on page 2 (see Footnote 2). The experiments in Section 5.2 examine this phenomenon through the lens of our novel theoretical results. Specifically, they show that wider neural networks yield lower mean and variance of our distributional distance $d_\mathrm{LLV}$ (second plot from the right in Figure 3), and that this correlates with lower values of our representational dissimilarity measure $d_\mathrm{SVD}$ (rightmost plot in Figure 3)—both in the extreme regime where the bound from Theorem 4.6 is non-vacuous (thereby illustrating our theory) and, intriguingly, even beyond that regime, suggesting that a tighter bound may be attainable. These results are inherently novel, as they rely on our newly introduced dissimilarity measure $d_\mathrm{SVD}$ and distributional distance $d_\mathrm{LLV}$. Importantly, prior experimental studies have used different similarity measures—most notably those based on CCA. However, similarity under CCA does not imply similarity under $d_\mathrm{SVD}$, since CCA is invariant to arbitrary linear transformations, whereas $d_\mathrm{SVD}$ is only invariant to a restricted class of such transformations (see Appendix E.8 and our response to reviewer Hbxo).

“Empirical characterization”

Our primary goal is to advance the mathematical understanding of representational similarity through the lens of identifiability. We appreciate that the reviewer considers lack of largeer scale experiments a “minor weakness,” and we agree that a more comprehensive empirical investigation is a valuable direction for future work. At the same time, we are encouraged by reviewer r8r8’s assessment that, “while not large scale, [our experiments] are well-designed to illustrate [our] argument.” In addition, we have conducted further experiments to strengthen the empirical validation of our approach; please see our response to reviewer 66wY for details.

“offering a tool should ideally be grounded in [...] application”

While our current work is primarily theoretical and only represents an initial step towards applications, we believe it is a significant and original one; in particular, it relaxes a strongly unrealistic assumption underlying many prior identifiability analyses—namely, the assumption of equal model distributions. We see this as an important step toward developing methods with empirical relevance and applicability. Encouragingly, several reviewers offered concrete suggestions for how this line of work could evolve toward practical applications (e.g., for distillation, as suggested by reviewer r8r8). We take this as a sign that our theoretical contribution opens up a promising avenue for future development with real-world impact.

[1] Hyvärinen, A., & Pajunen, P. (1999). Nonlinear independent component analysis: Existence and uniqueness results. Neural networks, 12(3), 429-439.