Intrinsic Dimension Correlation: uncovering nonlinear connections in multimodal representations

We thank the Reviewer for carefully evaluating and appreciating our work. We try here to address the weaknesses that emerged in the review.

W1 and Q1: In its current form, our method can only be applied to pairs of modalities. However, we are developing an extension that accommodates more than two modalities by leveraging the concept of Total Correlation [a], a multivariable generalization of Mutual Information, where entire modalities are considered instead of individual coordinates. However, this extension is still a work in progress and not yet ready for publication. If the Reviewer deems it appropriate, we are willing to remove references to this possibility in the final version of our paper.

W2: We agree that it would be helpful to have an intuitive graphical comparison between different correlation methods. However, we could not find a suitable graphical representation of the methods we use as baselines, as these offer a less immediate geometrical interpretation than intrinsic dimensionality.

W3: We are currently evaluating an application of $I_d$Cor for Cross-Modal Knowledge Distillation [b], in which knowledge is transferred across modalities from a large pre-trained teacher to a smaller student model. $I_d$Cor is a differentiable measure (as long as the $I_d$ estimator is differentiable, see [c] for a recent proposal in this field), and it can be employed as part of the training loss to enforce consistency between the two models. In particular, this setup could be useful in settings with limited data in one of the modalities, where training from scratch becomes infeasible.

Minor concerns: We thank the Reviewer for their suggestions. We reformulated the expressions to avoid using the Saxon genitive and inverted the panel ordering in figures 3 and 4.

[a] Satosi Watanabe, Information theoretical analysis of multivariate correlation. IBM Journal of Research and Development, 1960

[b] Zihui Xue, Zhengqi Gao, Sucheng Ren, and Hang Zhao, The modality focusing hypothesis: Towards understanding crossmodal knowledge distillation. ICLR, 2023

[c] Hamidreza Kamkari, Brendan Leigh Ross, Rasa Hosseinzadeh, Jesse C Cresswell, and Gabriel Loaiza-Ganem, A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models. NeurIPS, 2024

We are grateful to the Reviewer for their thoughtful feedback and for appreciating our work. We try to respond below to the weaknesses and questions that were presented in the review.

On the difference between $I_d$Cor and $I_d$: We thank the Reviewer for suggesting to clarify this important point: while the difference between the embedding dimension and the intrinsic dimension (alone) is a signature of the importance of the correlations within a dataset, it cannot be used for identifying correlations between different datasets, since the intrinsic dimension of the merged dataset would be affected both by the correlations within each dataset as well as those between the datasets. We now include this explanation in the introduction of our revised manuscript.

On the first bullet point: We reformulated this bullet point in the revised version of our paper.

On coarse alignment in self-supervised models: We thank the Reviewer for raising this point, as we were also intrigued by these results. We ran additional tests on another self-supervised vision-language model (VLM), BLIP [a], which has a similar architecture to CLIP but differs slightly in pre-training objective and data. As we report in our revised PDF, on this model $I_d$Cor produces a similar score to the one found in CLIP and SigLIP, even though in this case it is not surpassed by other metrics such as CKA and dCor. This consistency suggests that the pre-training objective plays a role in shaping the outcomes of our correlation metrics. This aligns with recent findings [b], which report significantly different representation similarity patterns in supervised vision models and VLMs. However, we do not see this as a limitation of our method, as correlation is still found with high confidence, as reported by the $p$-values, consistently at 0.01.

[a] Junnan Li, Dongxu Li, Caiming Xiong, and Steven Hoi, Blip: Bootstrapping language-image pre-training for unified vision-language understanding and generation. ICML, 2022

[b] Laure Ciernik, Lorenz Linhardt, Marco Morik, Jonas Dippel, Simon Kornblith, and Lukas Muttenthaler, Training objective drives the consistency of representational similarity across datasets. arXiv:2411.05561, 2024

We thank the Reviewer for carefully evaluating our work and for appreciating the idea behind $I_d$Cor. We respond below to the weak points identified in the review.

W1 and W2: We provide two additional experiments that we hope can clarify the discriminative power of $I_d$Cor, whose results are reported in the revised version of our manuscript in figures 12 and 13. In summary, in the first experiment we assess the correlation between image representations and multiple aligned textual representations, and show that $I_d$Cor successfully captures the decreasing strength of correlation as we progressively disrupt alignment between the two modalities. In the second experiment, we evaluate the correlation between visual representations computed on different image datasets, showing that in this case $I_d$Cor scores drastically decrease, while $p$-values increase. Regarding the experiment proposed by the Reviewer, from our understanding, we do not anticipate it being successful. The reason is that if the comparison involves distinct sets of image representations (even those with the same label), the $I_d$Cor score might reflect similarity but not true correlation. This is because there would be no correct alignment, even approximate, between the images, leading to high $p$-values and an unreliable score. However, if we have misunderstood any aspect of the proposed experiment, we are happy to participate in further discussions.

W3: We agree that baseline methods exhibit higher variance between models compared to $I_d$Cor, and we added a sentence that mentions this in the main text. However, the main purpose of figure 3 is to provide a first large-scale evaluation (both in terms of data and variety of encoders) for $I_d$Cor, in a relatively controlled and well-known unimodal setting, in which baseline methods are known to perform reasonably well.

W4: The Reviewer is right in saying that different metrics are under different scales. This is particularly evident as CKA and Distance Correlation produce scores well below 1 even in the linear case, due to their lack of invariance to invertible linear transformations ([a],[b]). However, the crucial point of this experiment is to show that, as the nonlinearity of correlation becomes more marked, baseline methods lose more signal, even in relative terms. In fact, while on average $I_d$Cor passes from 0.96 to 0.73 (losing 24% of the initial score), the closest performing baseline method, RBF kernel CKA, passes from 0.81 to 0.46, which represents a decrease of 43%.

[a] Simon Kornblith, Mohammad Norouzi, Honglak Lee, and Geoffrey Hinton. Similarity of neural network representations revisited. ICML, 2019

[b] Max Klabunde, Tobias Schumacher, Markus Strohmaier, and Florian Lemmerich. Similarity of neural network models: A survey of functional and representational measures. arXiv:2305.06329, 2023

Thanks. The author addressed my concern and provide additional experiment to support its claim. I would like to raise my score. And recommending accept this paper.

We thank the Reviewer for their feedback on our work, and respond here to the weaknesses and questions outlined in the review.

W1: The invariances of the $I_d$Cor method strongly depend on the intrinsic dimension estimator employed. When using the TwoNN estimator, the method is inherently invariant to rotation, translation, and global scaling. Additionally, due to its connection with mutual information, we expect $I_d$Cor to exhibit a significant degree of invariance to nonlinear transformations. This expectation is supported by Figure 2, which tests invariance as a function of the nonlinearity of the transformation. Although we have not specifically tested subset translation, it is likely that, as far as the points of the subset are clustered in the same region, this should not impact the TwoNN estimate, which relies on distances between the two nearest neighbors.

Regarding the tests proposed by Ding et al. [a], we do not anticipate meaningful results when applying them to $I_d$Cor. These tests are designed to evaluate indices that measure geometric similarities between representations, whereas our method quantifies the amount of shared information among representations. This fundamental difference affects the applicability and relevance of such tests. For instance, the sensitivity test would not be effective, as all principal components are linear combinations of the original coordinates and therefore retain their information by definition.

As noted in the last paragraph of section 4.2, our method focuses on detecting statistical dependencies between different representations rather than measuring geometric similarities. This does not imply that the index always yields a high value; rather, when measuring similarities between transformations, it is only sensitive to those that disrupt the flow of information, such as random shuffling of pairings.

In the revised version, we propose a new test to further illustrate this point. Specifically, we use a dataset of images with five possible captions to investigate whether the representations of this multimodal dataset are informative about one another. We then disrupt the correlations between captions and images by progressively shuffling the pairings and observe the following (see figure 12 in the revised manuscript): 1) a high correlation score when captions are correctly paired, 2) a gradual decrease in correlation as the pairings are partially shuffled, and 3) a low correlation score with high p-values, indicating low significance, when pairings are completely randomized.

It is worth noting that in these experiments, the $I_d$ of the merged dataset ranges from 40 to 80. This suggests that our method may encounter limitations due to the constraints of the intrinsic dimension estimator. Nonetheless, the observed trends align well with our theoretical expectations.

W2: We thank the reviewer for pointing us to these works. We computed the divergence scores using the Representation Topology Divergence (RTD) method [b] for the multimodal N24News representations we consider in figure 4 of our paper. Results showcase a clear disparity between modalities: excluding the diagonal (comparison of a model with itself), the average divergence among image encoders is 26.8, while the average scores for text encoders and across modalities are respectively 45.2 and 45. These results suggest that RTD is capturing a much stronger similarity in the visual modality than in the textual and multimodal settings. We did not include the full results in the current version of the manuscript because they are not directly comparable with $I_d$Cor, nor with the other baselines. In fact, RTD produces an unbounded divergence score, and not a correlation score, and we did not find any obvious way of scaling the results in a comparable range (simple min-max scaling would not be fair as it would imply zeroing-out a non null result). However, we are open to reconsider this choice in the final version of the paper if the Reviewer deems it appropriate.

W3: We agree with the Reviewer that the positioning of $I_d$Cor in this respect should be made more clear. We take the findings of [c] as a starting point for our investigation. In fact, while they observe that multimodal representations can be functionally similar (affine transformations are sufficient to map one modality into the other with satisfactory performance on a downstream task), which should imply some degree of correlation between those, we notice that current correlation (structural) methods widely employed to analyze neural representations capture weak to no correlation in the multimodal setting. This motivates us to propose an information-based correlation method to overcome the limitations of existing methods. We refactored the introduction of our paper to clarify this point.

We thank the Reviewer again for their valuable suggestions and proposed improvements to our work.

We evaluated $I_d$Cor in the suggested synthetic setting by generating a dataset of 3000 points organized into clusters of three. To create the dataset, we first sampled 1000 "anchor" points uniformly from the surface of the unit sphere. Each anchor point was then perturbed with three independent samples of Gaussian noise (mean $0$, standard deviation $0.01$) to form its associated cluster. Finally, we used $I_d$Cor to measure the correlation between this dataset and a modified version in which the clusters were randomly shuffled. $I_d$Cor with the TwoNN estimator is able to capture a decrease in correlation with respect to the non-shuffled case, reporting a coefficient of $0.62$ ($p$-value $0.01$). As one could expect in this setting, when using a "less-local" intrinsic dimension estimator, such as MLE with $k=100$ (number of neighbors), $I_d$Cor detects a more pronounced drop in correlation, yielding a coefficient of $0.29$ ($p$-value $0.01$).

We thank the Reviewer for pointing us to the work by MacKay and Ghahramani on the Maximum Likelihood Estimator. We will consider this improved estimator for future developments of our work.

Following the Reviewer’s suggestion, the new revision of our manuscript contains the full results of RTD on multimodal N24News representations in the Appendix.

We thank the Reviewer for their insightful feedback on our work and for suggesting several potential improvements to it. We try to address here the main weaknesses and questions that emerged in the review.

Choice of $I_d$ estimator: We agree that this is a crucial point, as our method fundamentally relies on $I_d$ estimation. We did experiment with Levina and Bickel’s MLE estimator [a], which is in fact implemented in the attached code in utils/intrinsic_dimension.py, but did not find outstanding differences with respect to TwoNN (results are similar or slightly less accurate in the controlled synthetic settings). Moreover, MLE has the disadvantage that it requires careful tuning of a hyperparameter, the number of nearest neighbors to consider. However, we provide results using MLE in the same synthetic settings of tables 1 and 2 in the Appendix (section A.3.1) of our revised PDF. Regarding the $I_d$ estimators employed by previous works on neural representations, references [b-e] only used TwoNN, [f] used both TwoNN and MLE, while [g] used a dozen of estimators, including TwoNN and MLE.

$I_d$ assumptions: We agree that the assumption of a uniform intrinsic dimension value over the entire dataset may not hold universally across all types of datasets. A potential solution, which requires the ability to reliably estimate $I_d$ locally, is to calculate the correlation score locally as well. This would allow for the computation of an $I_d$Cor score at the cluster level (by using Hidalgo [h]) or even at the individual point level (using IAN [i]). We have now clarified this point in the discussion section and included a proof-of-concept illustration of local $I_d$Cor in the Appendix. Moreover, we would like to thank the Reviewer for presenting to us the IAN method, which we were not aware of.

Synthetic results: We thank the Reviewer for raising these points. Yes, even though $p$-value computation involves repeated estimations of $I_d$ on permuted data, this does not involve resampling data. We now provide a more detailed evaluation in the proposed synthetic settings: all results are reported in terms of mean$\pm$standard deviation over multiple independent random samplings of data (with the exception of $p$-values, that are expressed in aggregate form as $[min, max]$ ranges) and we provide in the Appendix results with a different estimator (in section A.3.1, see the first weakness), with a different data sampling distribution (A.3.2), and when noise of increasing magnitude is applied to data (A.3.3). Moreover, we moved the main text section on synthetic experiments in the Results section, following the Reviewer’s suggestion.

Figure 2: We updated this figure in the revised version of our manuscript to include results from 10 independent random initializations of the MLP.

Figure 3: We followed this suggestion and added an experiment on correlating visual representations produced by the same model (CLIP-ViT-B) on different input datasets. The results are reported in figure 13 of the revised manuscript and they show that, as one would expect, $I_d$Cor score significantly drops when applied to different and uncorrelated representations, while the $p$-values increase. Moreover, in a similar spirit, we produced another experiment on multimodal Flickr30k representations, whose results are reported in figure 12. In this test, we pair each image representation with 5 caption representations, and show that $I_d$Cor consistently decreases as we progressively break the correlation between visual and textual representations.

Minor points:

• We adjusted the terminology to remove the source of confusion about 2D datasets.
• We agree that it would be interesting to accommodate a more detailed explanation of the positive impact that representation similarity analysis can have on interpretability, which is now limited to the second paragraph of section 2.1. However, due to space constraints, we are unable to extend our review.
• We completely agree with the Reviewer. The TwoNN method is now described in the Appendix.
• We reformulated the sentence in an attempt to clarify the point on $I_d$ estimation with PCA.
• We agree with the Reviewer and therefore we removed the references to dimensionality reduction methods.

Q1: As anticipated above, we included an analysis of the robustness of $I_d$Cor to noise in the Appendix of our revised manuscript, in section A.3.3.

Q2: Following the suggestion, we included $I_d\oplus$ in table 2.

Thank you very much for your comment. We were not aware of these papers and, therefore, we modified the last version of the paper accordingly .