Paper: Estimating Shape Distances on Neural Representations with Limited Samples (ICLR-2024)

Reviewer comment (Experiments):
One small experiment with VAEs (even if it is confined to the appendix) might be useful for quantifying the performance of stochastic networks.

Author response:
We have now included an experiment where we compare the representation of two neural networks trained on the same task but from different initialization points and with different training procedures (Appendix E: Applications to deep learning). Interestingly we find that the plug-in estimator, similarly to simulations, shows a high bias that slowly converges as we increase the number of samples, but the moment estimator maintains little bias regardless of the number of samples (Fig 5). We discuss how the ways in which the plug-in estimator’s bias can depend on irrelevant nuisance variables including number of samples and effective dimensionality may lead to erroneous scientific conclusions about the similarity between networks.

Reviewer comment (Experiments):
The overall motivation of the problem is a bit weak. The experiments don't clearly illustrate the downstream benefits of this work. Further experiments showing tangible gains on at least one real world application would greatly strengthen the paper in my opinion.

Author response:
We thank the reviewer for their thoughtful comments about our work, in particular for highlighting our novel theoretical analysis of this problem.  

We disagree with the reviewer’s comment that “experiments showing tangible gains on at least one real world application would greatly strengthen the paper” since our original submission did include an application to experimental neuroscience where data are noisy and sample-limited. Nonetheless we  acknowledge that the importance of studying the uncertainty in estimating shape distances may not have come across clearly in our introduction.  We have added paragraph 3 in the introduction to make clear that, although the Procrustes and shape distances brought to light by Williams et. al. (2021) are currently being applied in this field to measure representational similarity between both artificial and biological neural networks (see: Schuessler et. al. 2023 (https://arxiv.org/abs/2307.07654), Lange et. al. 2022 (https://arxiv.org/abs/2206.10999), Boix-Adsera et. al. 2022 (https://arxiv.org/abs/2210.06545), Alvarez-Melis et. al. 2018 (https://arxiv.org/abs/1806.09277), and Ding et. al. 2021 (https://arxiv.org/abs/2108.01661)), there has been little progress in understanding the theoretical and empirical behavior of the estimators for these distances.  This is a major gap in understanding, because—as we show—the naive plug-in estimators are biased and can converge slowly, and there are circumstances when less naive estimators such as the estimator we propose can be advantageous.  For instance, Figure 2B shows us that in the fixed dimensionality case, one may reduce the variance of an essentially unbiased estimate by increasing the number of stimuli sampled, faster than the bias of the plug-in estimator can be reduced in the same way.  

In our additional experiments on an artificial neural network (introduced in the rebuttal period), we can see that the bias of the naive plug-in converges slowly with respect to increasing samples whereas the moment estimator maintains small bias (Fig 5, Appendix E).  These findings are relevant to the active research being performed to compare representations in neural networks using Procrustes/shape distances (see citations in previous paragraph). Furthermore, the bias of the plug-in estimator may depend on nuisance properties of the populations being compared leading to confounds. We concretely demonstrate this by showing that the bias of the plug-in estimator depends on the effective dimensionality of the neural populations (Fig 6, Appendix E). Thus plug-in estimated differences across experimental conditions could simply result from differences in effective dimensionality and not the actual similarity. The quantification of bias in the moment estimator allows the practitioner to precisely determine if differences cannot be explained by estimator bias – a highly practical scientific benefit to the novel estimator.

Reviewer comment (Evaluation):
In Section 4.2 it is acknowledged that the proposed estimator is highly variable in low SNR regimes and so neurons with the highest SNR are selected. Can you provide the relative proportion of such neurons in the presented scenario and also (if possible) comment on their prevalence in general? This is important because if the relative proportion is small then it means that the proposed estimator can only be applied in very few cases.

Author response:
These 40 neurons are the maximal SNR units so they are exceedingly rare, 40 out of 10,000 neurons recorded. This is a wide-field calcium imaging experiment— known to be noisy. These results suggest that accurate shape metric estimation in large neural populations will require higher SNR, potentially with electrophysiological recordings. The confidence intervals of our moment estimator quantify this whereas the plug-in estimator naively interpreted would have led to biased results.