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Spectral Analysis of Representational Similarity with Limited Neurons
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Uses random matrix theory to derive analytical formulas for how neuron subsampling distorts CCA/CKA via eigenvector delocalization, and proposes a constrained-optimization denoiser that infers population similarity from power-law spectra.
Problem
Representational similarity measures (CCA, CKA) computed from a limited number of recorded neurons are biased, typically underestimating population-level similarity between neural data and models.
Approach
Relates sample CCA/CKA to population spectra via random matrix theory overlap formulas (self-overlap and cross-overlap of eigenvectors), attributes underestimation to eigenvector delocalization, and inverts the forward overlap relation with a power-law eigenvalue prior to denoise population similarity.
Key results
- Sample CKA is systematically underestimated under neuron subsampling due to eigenvector delocalization; with N=30 and P=200 the sample CCA ranking of two models flips relative to population CCA.
- For power-law spectra the number of localized eigenvectors scales as (1+γ)/√8·√N, so reliably resolving k components needs N ≳ 8k²/(1+γ)².
- On primate V2 data with N=20 of 103 neurons, the inferred population similarity is consistently higher than naive sample similarity.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| limit neural recordings to fewer neurons | neural representations compared with model representations · Synthetic and primate visual cortex data | CKA | Hurt |
| limit neural recordings to fewer neurons | neural representations compared with model representations · Synthetic data | CCA | Hurt |
| infer population similarity using the denoising framework | sample-based model–brain similarity · Synthetic data | CKA | Helped |
| infer population similarity using the denoising framework | sample-based model–brain similarity · Synthetic data | CCA | Helped |
| infer population similarity using the denoising framework | sample-based model–brain similarity · Primate visual cortex V2 | CKA | Helped |
| infer population similarity using the denoising framework | sample-based model–brain similarity · Primate visual cortex V2 | CCA | Helped |
| sample neurons rather than use the full population | model comparisons with the brain · Synthetic data | CCA ranking | Mixed |
| sample a limited number of neurons from a power-law population spectrum | population eigenvector recovery · Power-law spectra | number of localized eigenvectors: i⋆(N) ≈ (1+γ)/√8 √N | Hurt |
Surprising findings
- Sampling neurons can reverse the similarity ranking of two models relative to the brain even though population similarities are ordered the other way.
- Sample eigenvector overlaps do not self-average even as P grows, so trial averaging or bootstrapping is needed.
Limitations
- Assumes Gaussian linear random projections of latent population codes; nonlinear and non-Gaussian data may reduce accuracy.
- Models only rotational invariance; does not cover dynamic or nonlinear similarity metrics.
- Denoising recovers only the leading localized eigencomponents and assumes sufficiently large datasets for negligible bias.
Experimental setup
Datasets: CIFAR-10 primate V2 cortex recordings
Benchmarks: synthetic population Gram matrices primate V2 neural recordings
Metrics: CKA CCA SVCCA self-overlap cross-overlap
Baselines: moment-based CKA estimator naive sample CKA naive sample CCA
Builds on: Random matrix theory overlap formulas Baik-Ben Arous-Péché phase transition Silverstein equation SVCCA Kornblith CKA
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