Title: Spectral Analysis of Representational Similarity with Limited Neurons
Venue: NeurIPS 2025
Accepted: yes
Keywords: representational similarity

centered kernel alignment

random matrix theory

computational neuroscience
Abstract: Understanding representational similarity between neural recordings and computational models is essential for neuroscience, yet remains challenging to measure reliably due to the constraints on the number of neurons that can be recorded simultaneously. In this work, we apply tools from Random Matrix Theory to investigate how such limitations affect similarity measures, focusing on Centered Kernel Alignment (CKA) and Canonical Correlation Analysis (CCA). We propose an analytical framework for representational similarity analysis that relates measured similarities to the spectral properties of the underlying representations. We demonstrate that neural similarities are systematically underestimated under finite neuron sampling, mainly due to eigenvector delocalization. To overcome this, we introduce a denoising method to infer population-level similarity, enabling accurate analysis even with small neuron samples. Theoretical predictions are validated on synthetic and real datasets, offering practical strategies for interpreting neural data under finite sampling constraints.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies how sampling a subset of neurons in a large population (modeled as Gaussian projection) affects estimates of the alignment between the sampled representation and some fixed reference population. Its main result applies existing characterizations of eigenvector overlaps of Wishart matrices to describe how eigenvector delocalization induces bias in plug-in estimators of representational alignment. The authors then leverage this characterization to develop a corrected estimator.\n\nSTRENGTHS AND WEAKNESSES:\nOn the whole, I think that this paper is timely, and will certainly be of interest to the community of researchers working on representational alignment. In my view, its main weakness is that the connection between the theoretical results and the proposed algorithm for obtaining corrected estimates of alignment is not so strong. For instance, so far as I can tell the authors do not have a guarantee that it produces an asymptotically unbiased estimate. This is an important limitation because the authors make quite a bit of the fact that their approach comes with more theoretical insight relative to unbiased estimators based on moment computations (Line 233). Most of my questions below are focused on this point. I also have a few suggestions for how the authors could improve the clarity of their manuscript, but those are minor concerns in comparison.\n\nQUESTIONS:\n1. The main technical limitation I see in this paper is that the authors do not provide a full characterization of the bias of their proposed algorithm. The justification provided in 4.2.2 for why it works is heuristic in nature. It would be useful if, even just for the setting of power law data, you could give a more complete theoretical analysis of when the algorithm works. For instance, fixing a particular population decay exponent, how many components does one have to accurately estimate in order to estimate the CKA/CCA within some desired error tolerance, and when can the constrained optimization used to estimate $\\tilde{M}$ succeed in estimating that number of components? How does the choice of algorithms used to estimate the power law exponent and the overlap matrix affect the final estimate of the CKA/CCA? The lack of clarity around this issue is my main reason for recommending rejection. \n\n2. The authors mention moment-based unbiased estimators for the CKA in lines 231-235, but do not compare these methods to their proposed estimator. Adding a few such comparisons would help the reader choose a method. In particular, it would be useful to include those alternative estimators in Figure 4.\n\n3. The series of approximations made in Appendix C is not clearly motivated. Can you elaborate on why each of the assumptions made in order to render the computation tractable make sense in the context of the neuroscience data of interest? \n\n4. I have several formatting suggestions, which I think would clarify the presentation:\n\n    1. I think it would be helpful if the authors stated their main theoretical result (deterministic equivalents for CCA and CKA) as a proposition or theorem, since as written the lede is somewhat buried in Section 3 (the main result is stated in Lines 134-135 as \"Plugging in the formula for expected $M_{ia}$ in Eq. 5 and Eq. 6, we get an analytical formula for CCA and CKA, respectively\"). \n\n    2. The paper employs somewhat non-standard notation for sample and population covariance matrices, denoting them by $\\Sigma$ and $\\tilde{\\Sigma}$, respectively. Unless the authors have a good reason for not doing so, I suggest that they adopt the more standard notation of $\\hat{\\Sigma}$ and $\\Sigma$, where quantities estimated from samples are distinguished from their population counterparts by a circumflex.\n\nLIMITATIONS:\nOn the whole, I think there is room for the authors to better address the limitations of their work. One point I wish they remarked upon more clearly is the fact that they do not consider the effect of random stimulus sampling; a sentence or two in the discussion would suffice. Some more prose could also be devoted to the limitations of the proposed algorithm.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-20"}

comments: SUMMARY:
This paper demonstrates that in certain theoretically tractable settings, subsampling neural populations systematically underestimates neural similarity.

STRENGTHS AND WEAKNESSES:
This paper tackles an important question at a good time. It is mostly well-written, and the motivation is very clear. 

The main weakness is that it really helps to be a physicist when reading this paper, which I am not (I am coming from dynamical systems / control theory / ML).

QUESTIONS:
1. From line 89, it appears that subsampling is modeled using a random projection matrix $R$. However, it's not immediately clear to me why this is an appropriate model for subsampling. Intuitively, I would expect subsampling to involve selecting specific units rather than mixing information across all units, as a random Gaussian projection does. Could you clarify the rationale behind this modeling choice?
2. In the literature, one typically encounters similarity metrics which are normalized by the maximum explainable variance (i.e., similarity is measured relative to a noise ceiling). See, for example, equation 1 [here](https://arxiv.org/pdf/2502.16238). If you were to use this metric in your setting, would you still be systemically underestimating the neural similarity? 
3. There has been much recently work on similarity metrics between dynamical systems, starting with [DSA](https://arxiv.org/pdf/2306.10168) but see also [this](https://arxiv.org/abs/2410.24070) work. Do you think your theoretical insights would carry over into this setting as well? If that is too difficult to answer, do you think Procrustes-based similarity metrics also suffer from this systematic bias? It would be nice to see some discussion on this front.

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 3
IMPACT: 4
ORIGINALITY: 3
DATE: 2025-06-22

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper analyze the representation similarity in neural networks with limited neurons, which is a common scenario in neuroscience as the recorded neurons are often limited in number. The authors develop a theoretical framework based on Random Matrix Theory to analyze how finite neuron sampling systematically underestimates similarity measures like CCA and CKA. They propose a denoising method to infer population-level similarity from limited samples and validate their approach on both synthetic and real neural data.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n1. The research question of this paper is important.\n2. The authors develop a theoretical framework based on Random Matrix Theory to analyze the effects of limited neuron sampling on similarity measures, which is a novel approach.\n\n## Weaknesses\n**Minor concerns**\n1. My primary concern is that applying CCA and CKA to analyze neural representational similarity may be fundamentally inappropriate, as these `representational metrics completely ignore the temporal dynamics of neural activity`. This limitation has been extensively discussed in recent similarity measures literature [1-3], highlighting two critical issues:\n- Different dynamical systems can yield similar static representations when sampled at discrete time points, and\n- Similar underlying dynamics can manifest as different static representations depending on sampling timing.\n\n    However, both CCA and CKA are incapable of addressing these two scenarios, making the comparison between neural recording potentially misleading.\n\n2. The scalability of the authors' approach can be constrained by its dependence on the spectral decomposition properties specific to CCA/CKA. The method cannot easily extend to:\n- Dynamic similarity metrics\n- Non-linear similarity metrics\n- Information-theoretic approaches\n  This raises concerns about the broader impact of this work: if the field is moving away from `representation similarity metrics` toward other directions above, then developing sophisticated corrections for potentially obsolete metrics may have limited long-term value for computational neuroscience.\n\n3. The assumption that neuron sampling can be modelled as `random projections` is mathematically convenient but can be a dangerous oversimplification. In practice, electrode placement is often anatomically constrained, and recorded neurons may be spatially clustered or biased toward certain sub-regions. This could violate the independence assumptions underlying the random matrix theory.\n\n**Minor issues**\n1. The notation switches between tildes and hats for population vs. estimated quantities, which could be clarified for consistency.\n\n**Reference**\n1. Ostrow, M., Eisen, A., Kozachkov, L., & Fiete, I. (2023). Beyond geometry: Comparing the temporal structure of computation in neural circuits with dynamical similarity analysis. Advances in Neural Information Processing Systems, 36, 33824-33837.\n2. Kamiya, S., Kitazono, J., & Oizumi, M. (2024, March). Koopman Operator Based Dynamical Similarity Analysis for Data-driven Quantification of Distance between Dynamics. In ICLR 2024 Workshop on Representational Alignment.\n3. Zhang, S., Ye, Z., Yan, Y., Song, Z., Wu, Y., & Wu, J. (2025). KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling. In Forty-second International Conference on Machine Learning.\n\nQUESTIONS:\nSee the `Weaknesses` section.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 1, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nMeasuring representational similarity between computational models and neural recordings has become a staple methodology in computational neuroscience. In practice, although we can access the entire population of neurons in computational models, we can record only a limited number of neurons in the brain, hoping they satisfactorily represent the larger population. In this work, the authors apply Random Matrix Theory to investigate how this sampling limitation impacts similarity measures. They find that similarity scores from popular measures are systematically underestimated under finite neuron sampling, and they propose a method to infer the population similarity. Theoretical predictions are validated on both synthetic and real datasets.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths \n- This paper tackles an important problem and contributes a practically useful methodology for the community. Representational similarity metrics are broadly used to infer good computational models of the brain, and we need to be able to trust those measures despite the practical limitations when recording neural activities.\n- The paper is well written and well illustrated.\n- The investigation carried out in the paper is thorough and convincing. \n\n### Weaknesses\n- I do not see any major weakness in this paper.\n\nQUESTIONS:\n- Fig. 2: How many sampling is done per neuron size N?\n- Why is figure 5 used before figure 4 in the text?\n- The proof that CKA or CCA are underestimated differently across models is strong enough evidence to motivate the usefulness of the proposed correction. That being said, it would have further strengthened the paper to push further the empirical investigation in section 5.2 to see if any of the previously established insights change when applying the proposed correction.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}