Thank for your thoughtful review. We are very pleased that you found the paper a pleasure to read!

• We plan to apply our method to neural data and are actively doing so in ongoing work (see our general response Algorithmic complexity and evaluation on neural data).
• We are working on including the important aspects of the derivation while staying within the 10 page limit.
• We will improve the introduction to make the motivation more clear. We give a brief review of relevant literature below.
  • Artificial deep networks perform computations by cascaded manipulations of vector-valued input vectors (“representations”) and this is thought to be a reasonable approximation to what happens in biological systems. The geometry of these hidden layer representations ought to give us hints as to how the systems work as well as their vulnerabilities (e.g. Kriegeskorte & Kievit, 2013; Trends in cognitive sciences), and a large body of work leveraged representational similarity measures like centered kernel alignment (CKA; Kornblith et al. 2019), canonical correlations analysis (CCA; Raghu et al. 2017), and Procrustes distance (Williams et al., 2021) to compare the geometry of neural representations across different artificial and biological systems. These distance measures can be used for a variety of things like comparing whether deep vs. wide neural networks utilize similar representations (Nguyen et al., 2020). Another example is the recently proposed “Platonic Representation Hypothesis” (Huh et al., 2024), which suggests that deep networks converge onto the same representation when trained on very large datasets. Quantifying this hypothesis will require distance metrics between systems.
  • Our paper extends this body of work to handle stochastic and dynamic representations. Duong et al. (2023) recently proposed a metric for stochastic, non-dynamic neural representations, and Ostrow et al. (2023) recently proposed a metric for dynamic, non-stochastic representations. Both were published at high quality venues, ICLR and NeurIPS, and are well cited. Ours is an important step forward because it leverages both dynamic and stochastic structure to form a distance measure. Figure 1 in our paper aims to illustrate the advantages of this.

Thank you for your review. We are very pleased to hear that you found that paper well-written, well-motivated at timely.

We agree that including a comparison with DSA is important and have added these comparisons (see our general response).

Thank for your careful reading of our paper and for your thoughtful comments. We have addressed your concerns about the Gaussian process assumption, the computational inefficiency of our algorithm and the comparison to DSA in our general response.

In response to your questions:

• Importantly, our metric only depends on first- and second-order statistics of processes ${\bf x}$ and ${\bf y}$. Therefore, increasing the deviation in the higher-order moments between the processes ${\bf x}$ and ${\bf y}$ will not affect the Causal OT distance between ${\bf x}$ and ${\bf y}$ as long as their first- and second-order moments remain the same. We agree it would be interesting to develop methods that can capture differences in higher-order moments. One possibility (following Duong et al. ICLR, 2023) is to extend energy distances to compare stochastic processes.
• Indeed, this is an important point: our method is computationally intensive when applied to high-dimensional neural data. We are actively thinking about methods to improve the efficiency of our method when applied to high-dimensional neural data (see our general response Algorithmic complexity and evaluation on neural data).
• Our method does not make a low-dimensional assumption, so in general it will not suffer from the problems documented in Qian et al. 2024. We agree it would be interesting to test the ability of our method to differentiate line attractors versus feedforward amplification via non-normal dynamics.
• See our general response Comparison with DSA
• We are intent on applying our method to neural data. We view this as a methodological paper and plan to include neural data in follow up work (see our general response Algorithmic complexity and evaluation on neural data).

Dear Reviewer,

Thank you again for taking the time to provide feedback on our submission. We have responded to your concerns in our rebuttal and would greatly appreciate it if you could review our clarifications and let us know if there are any further points we can address before the discussion period ends. Thank you!

Gaussian process (GP) assumption

Reviewer 4ecC raised concerns about the assumption of Gaussianity, which does not hold for neural processes. We would like to make a few clarifications.

• While it is true that the GP is the running example in the paper, it’s worth clarifying that Causal OT is a pseudometric over all stochastic processes in the sense that $d({\bf x},{\bf y}) = 0$ if and only if the mean and covariance of ${\bf x}$ and ${\bf y}$ are equal. This follows easily from the Bures + adapted Bures distances being a proper metric on covariance. Thus, strictly speaking, none of the distances we compute assume a GP.
• If we do make a GP assumption, there are some additional nice benefits. The distances we propose are proper metrics over Gaussian processes in that if ${\bf x}$ and ${\bf y}$ are GPs then $d({\bf x},{\bf y}) = 0$ if and only if ${\bf x} ={\bf y}$. Additionally, the interpretation of the distance in terms of optimal transport holds only when ${\bf x}$ and ${\bf y}$ are GPs. But these are just nice bonus features.
• Causal OT distances that incorporate higher order moments are defined for arbitrary (GP or non-GP) stochastic processes (Backhoff et al. SIAM J. Opt., 2017). However, statistical estimators for these quantities suffer a severe curse of dimensionality and are generally viewed as intractable (e.g. Boissard & Le Gouic AIHP, 2014) .
• We agree with the reviewer that the distance we present in the paper (Causal OT on GPs) can fail to distinguish adversarial cases where the mean and covariance of the processes are tuned to be the same while higher order moments are different. However, despite their limitations, GPs are a popular framework for modeling neural dynamics (e.g. Duncker et al. ICML 2019).

Comparison with DSA

Reviewers 4ecC and Sz7A state that a comparison of Causal OT with DSA would strengthen our results. We now provide comparisons of our method with DSA on all of the experiments that we performed (except for the 2-step example since DSA requires delay-embedding the process). We also added 2 appendices that further explore the performance of DSA and how it compares with Causal OT (as well as with Procrustes and SSD). While our results are not comprehensive, we generally find the following hold:

• DSA does not distinguish between systems with the same underlying dynamics but different noise statistics (Appendix F). This is not surprising given that DSA was not designed to differentiate between systems with different noise statistics.
• DSA effectively distinguishes between systems in some examples (Figs. 2 and 4) but not all examples (Fig. 5), and the efficacy of DSA is sensitive to the choice of hyper-parameters (Appendix G).

Here, we provide a more detailed account of what we added:

• We include a description of DSA (section 2.3).
• We include a comparison with DSA in Figs. 2, 4 and 5.
  • Fig. 2: DSA distinguishes between all three systems. However, this required carefully choosing the hyperparameters (number of delays and rank). For other choices of hyperparameters, DSA fails to distinguish between the systems (Appendix G).
  • Fig 4: DSA distinguishes between the preparatory activity and the readout activity, even when preparatory activity is low. This is likely because changing the levels of preparatory activity does not change the topology of the system.
  • Fig. 5: DSA fails to distinguish between different conditional processes. Due to time constraints, we did not perform an exhaustive test of hyperparameter choices for DSA, so we cannot rule out the possibility that DSA would effectively distinguish the conditional processes using another choice of hyperparameters.
• We compare Procrustes, SSD, Causal OT and DSA on a scalar AR process with varying dynamics and noise levels (Appendix F). Procrustes fails to distinguish any of the systems (as expected since the systems are all mean zero). SSD distinguishes systems with the same dynamics but different noise levels; however, SSD does not distinguish systems with the same marginal noise statistics but different dynamics. DSA distinguishes systems with the same marginal noise statistics but different dynamics; however, DSA does not distinguish systems with the same dynamics but different noise levels. Causal OT distinguishes all of the systems.

Algorithmic complexity and evaluation on neural data

Reviewer 4ecC expressed concern about the computational complexity of the proposed method and Reviewers 4ecC and mGZm stated that they would like to see the method tested on neural data.

1. While the focus of this work has been laying the foundation for developing distances that respect stochasticity and dynamics in data, it is indeed critical to have algorithms for efficient computation of these distances. We discuss this in the Limitations section of the paper, and reference two related works that can enable efficient computation of Causal OT distance. Importantly, one of the two papers (Nejatbakhsh et al. NeurIPS 2023) explicitly models marginal covariances without incorporating across time covariances, while the other work (Duncker et al. ICML 2019) models the low-dimensional dynamics through latent variables with assumptions on the covariances.
2. In ongoing work, we’re combining these two techniques and developing a latent variable model that is flexible enough to capture dynamics and both marginal and across time covariances, which can then be used to estimate distances between neural processes. The generative model and inference are involved and require more space for exposition. We have applied the model to multiple neural datasets and estimated covariances; however, this process involves close collaboration with experimental neuroscientists and we believe this is beyond the scope of this methodological paper.