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arxiv: 2605.26973 · v1 · pith:COYRO74Jnew · submitted 2026-05-26 · 📊 stat.ML · cond-mat.dis-nn· cs.LG· cs.NE· q-bio.NC

Signal-to-Noise Ratio and Sample Size Govern Representational Alignment in Neural Networks

Ali Hussaini Umar , Alessandro Laio This is my paper

Pith reviewed 2026-06-29 15:44 UTC · model grok-4.3

classification 📊 stat.ML cond-mat.dis-nncs.LGcs.NEq-bio.NC
keywords representational alignmentsignal-to-noise ratiosample sizeinterpolation thresholdneural networksgeneralization errorlinear network
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The pith

Representational alignment in neural networks increases monotonically with signal-to-noise ratio but dips to a minimum near the interpolation threshold as sample size grows.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how similarly different neural networks represent data when each is trained on noisy versions of the same task. It perturbs training sets with independent noise realizations and measures alignment across ensembles of networks on both regression and classification problems. Alignment rises steadily as the signal-to-noise ratio improves. Alignment varies non-monotonically with the number of training examples, reaching its lowest point near the sample size at which the network can perfectly fit the training data. The same pattern appears in an exactly solvable linear network and in nonlinear networks on real data, and it is independent of generalization error.

Core claim

Across linear and nonlinear networks, regression and classification tasks, and both synthetic and real-world data, we consistently observe that alignment varies monotonically with SNR but non-monotonically with training sample size. In particular, the alignment is minimized near the interpolation threshold, and a stronger alignment does not necessarily correspond to better generalization error.

What carries the argument

Training ensembles on independently noise-perturbed versions of the same dataset, with analytic alignment computation in a single-hidden-layer linear network.

If this is right

  • Alignment increases as the signal-to-noise ratio of the training data increases.
  • Alignment reaches a minimum near the interpolation threshold when sample size is varied.
  • Stronger alignment does not imply lower generalization error.
  • The same dependence on SNR and sample size appears in both the linear analytic model and nonlinear networks on real data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Data collection strategies could target sample sizes away from the interpolation threshold if high alignment is needed for a downstream task.
  • The observed decoupling suggests alignment and generalization are controlled by distinct aspects of the training dynamics.
  • Non-monotonic dependence on sample size may appear in other similarity measures between networks.

Load-bearing premise

The specific noise process and alignment metric make the linear network's behavior representative of nonlinear networks trained on real data.

What would settle it

Measure alignment in networks trained at multiple sample sizes around the interpolation threshold and find that it does not reach a minimum there.

Figures

Figures reproduced from arXiv: 2605.26973 by Alessandro Laio, Ali Hussaini Umar.

Figure 1
Figure 1. Figure 1: Representational alignment depends on the signal-to-noise (SNR) ratio and increases non-monotonically with training sample size (n). Panel (a): presents the generalization error of a two-layer linear network as a function of training sample size divided by the network’s parameters for different SNR (at global minimum). The solid lines correspond to the theoretical predictions, while the dots with bars deno… view at source ↗
Figure 2
Figure 2. Figure 2: Empirical results for non-linear neural network. Panel (a): presents the average general￾ization error of a two-layer network with ReLU activation as a function of training sample size divided by the network’s parameters for different SNR values. Panel (b): presents the average Conditional Copula Entropy between hidden representations of identical networks trained on independent datasets. Numerical results… view at source ↗
Figure 3
Figure 3. Figure 3: Alignment between Representations of Linear and Nonlinear Network. Panel (a): presents the average generalization error of a two-layer network with linear (orange curve) and ReLU (blue curve) activation as a function of training sample size divided by the network’s parameters for SNR= 5. Panel (b): presents the average Conditional Copula Entropy between hidden representations of networks with Linear and Re… view at source ↗
Figure 4
Figure 4. Figure 4: Representational alignment of independently trained classification networks: Panel (a) presents the generalization error of the trained networks for various label noise. Panel (b) presents the CCE between penultimate representations of identical networks trained on independent datasets, starting from the same initialization. All curves are plotted as a function of the ratio between the training sample size… view at source ↗
Figure 5
Figure 5. Figure 5: Representational alignment of independently trained classification networks: Panel (a) presents the generalization error of the trained networks for various label noise. Panel (b) presents the CCE between penultimate representations of identical networks trained on independent datasets, starting from the same initialization. All curves are plotted as a function of the ratio between the training sample size… view at source ↗
read the original abstract

Neural networks are known to develop latent representations that are $aligned$, namely structurally similar across networks trained with different architectures, training protocols, or training datasets. We study this phenomenon in a controlled setting, where we train an ensemble of networks on regression and classification tasks using training sets perturbed by independent realizations of a noise process. We show that the signal-to-noise ratio (SNR) and the training sample size influence the alignment in qualitatively similar ways in networks trained on real-world datasets and in an extremely simple $linear$ network with a single hidden layer, for which the alignment can be estimated analytically. Across linear and nonlinear networks, regression and classification tasks, and both synthetic and real-world data, we consistently observe that alignment varies monotonically with SNR but non-monotonically with training sample size. In particular, the alignment is minimized near the interpolation threshold, and a stronger alignment does not necessarily correspond to better generalization error. These findings reveal a non-trivial dependence of alignment on data quality and quantity, decoupled from generalization performance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper studies representational alignment across an ensemble of neural networks trained on independently noise-perturbed copies of the same dataset. It claims that alignment increases monotonically with signal-to-noise ratio (SNR) while varying non-monotonically with training sample size (minimum near the interpolation threshold), that these patterns hold for both an analytically solvable linear network and nonlinear networks on synthetic and real data, and that alignment is decoupled from generalization error. The linear-network derivation supplies an independent check on the empirical observations.

Significance. If the central claims hold, the work supplies a concrete, analytically grounded account of when and why representations align across models. The closed-form linear-network result is a clear strength, as is the consistency of the SNR and sample-size patterns across regression/classification, synthetic/real data, and linear/nonlinear regimes. The decoupling from generalization performance is a useful negative result for interpretability and ensembling research.

minor comments (4)
  1. §3.2, Eq. (8): the alignment metric is defined via centered kernel alignment; a one-sentence reminder of its invariance properties would help readers connect it to the linear closed form.
  2. Figure 4 caption: the interpolation threshold is marked but the precise definition (e.g., number of parameters vs. effective degrees of freedom) is not restated; readers must hunt in §4.1.
  3. §5.3: the sensitivity analysis to noise-process parameters is present but reported only for two values of σ; a brief table or inset showing the monotonicity slope across a wider grid would be helpful.
  4. Table 2: the R² values for the linear-network fit are given, but the number of independent noise realizations used to compute each point is not stated in the table or caption.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the work and the recommendation of minor revision. The report does not list any specific major comments.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central results are observational comparisons: alignment is shown to vary monotonically with SNR and non-monotonically with sample size (minimum near interpolation) both in an analytically solvable linear network and in nonlinear networks on real data. The linear analytic estimate is derived independently from the model equations rather than fitted to the nonlinear results, and the alignment metric is defined explicitly without reducing to a parameter fit by construction. No self-citations, ansatzes smuggled via prior work, or renaming of known results appear in the provided abstract or reader summary. The derivation chain is self-contained against external benchmarks (analytic solution + controlled experiments), yielding a normal non-circular outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only; no free parameters, invented entities, or non-standard axioms are mentioned. The work relies on standard definitions of alignment and training dynamics.

axioms (1)
  • standard math Standard mathematical definitions of representational alignment and interpolation threshold apply to both linear and nonlinear networks.
    Invoked implicitly when claiming qualitative similarity between the analytically solvable linear case and real networks.

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