Research map / Learning theory and algorithms / Sample complexity
Estimating Shape Distances on Neural Representations with Limited Samples
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Derives upper and lower bounds on plug-in estimation error of shape distances (Procrustes and Riemannian) for neural representations and proposes a method-of-moments estimator with a user-tunable bias-variance tradeoff and approximate confidence intervals.
Problem
Estimating shape distances between high-dimensional, noisy neural representations from limited samples is biased and slowly converging, and existing work rarely quantifies estimator uncertainty.
Approach
Plug-in estimate of the nuclear norm of the cross-covariance is replaced by a polynomial expansion in eigenmoments W_p = Tr[(Sigma Sigma^T)^p], with coefficients chosen by a convex quadratic program that bounds maximal bias at a user-specified level while minimizing variance; moments are estimated unbiasedly and covariance is bootstrapped.
Key results
- Plug-in error bound scales as N M^{-1/2}, and a lower bound of (16B^2/3π) N^{1/2} M^{-1/2} shows the rate cannot be improved in M.
- On synthetic data, the moment estimator (constrained to 5% bias) keeps small bias at small M, while the plug-in estimator's bias explodes as N grows; 95% CIs miss the true value in only 2.3% of simulations.
- On neural data with true similarity 0, the moment estimator correctly reports near-zero similarity where the plug-in is upwardly biased; with true similarity 1 the moment estimator is biased low and its CIs span nearly the full range.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the moment-based estimator instead of the plug-in estimator | Shape similarity estimation · Synthetic simulations with low sample size and high dimensionality | bias | Helped |
| Use the moment-based estimator instead of the plug-in estimator | Shape similarity estimation · Synthetic simulations | variance | Hurt |
| Constrain the moment estimator to 5% bias | Moment-based estimator · Synthetic simulations with varying sample size | bias | Helped |
| Increase the sample size | Moment-based estimator · Synthetic simulations | variance | Helped |
| Increase representational dimensionality | Plug-in estimator · Synthetic simulations | bias | Hurt |
| Increase representational dimensionality | Moment-based estimator constrained to 10% bias · Synthetic simulations | variance | Hurt |
| Use the moment-based estimator instead of the plug-in estimator | Shape similarity estimation · Mouse V1 neural data with ground-truth similarity 0 | bias | Helped |
| Use the moment-based estimator instead of the plug-in estimator | Shape similarity estimation · Mouse V1 neural data with ground-truth similarity 1 | bias | Hurt |
| Use the moment-based estimator instead of the plug-in estimator | Shape similarity estimation · ResNet-50 ImageNet representations with small sample sizes | similarity estimation error | Helped |
| Use the moment-based estimator instead of the plug-in estimator | Shape similarity estimation · ResNet-50 ImageNet representations with small sample sizes | variance | Hurt |
| Compute approximate confidence intervals using estimated bias and variance | Moment-based estimator · Synthetic simulations | CI miss rate: 2.3% | Helped |
| Use all stimuli rather than a 400-stimulus fold | Moment-based estimator on mouse V1 data · Mouse V1 neural data | confidence interval width | Helped |
Surprising findings
- When true similarity is 1 on neural data, the moment estimator has worse bias than the plug-in estimator.
- On neural data with all stimuli, confidence intervals are wide, so even high-SNR neurons yield only intermediate similarity information.
Limitations
- Estimator variance grows substantially with dimension N and as bias bound is tightened; neural data with low SNR makes estimates highly variable.
- Analysis assumes bounded neural responses and i.i.d. inputs; noise is ignored by treating stochastic networks via replicates.
- Confidence intervals are conservative and can be nearly uninformative in noisy data.
Experimental setup
Datasets: ImageNet CIFAR-10 mouse V1 calcium imaging recordings (2,800 natural images)
Benchmarks: synthetic multivariate normal simulations mouse V1 calcium imaging neural data ResNet-50 penultimate-layer representations
Metrics: bias variance MSE confidence interval coverage
Baselines: plug-in estimator
Builds on: Procrustes shape distance Riemannian shape metrics matrix Bernstein inequality
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