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

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 and simplified 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 and typically non-monotonically with training sample size. In particular, the alignment is minimal near the interpolation threshold. In the linear theory, alignment collapses at interpolation in the ridgeless limit, whereas explicit regularization prevents this collapse and yields a nonzero floor, as observed in nonlinear networks; moreover, in the regularized linear model, the regularization strength that minimizes the generalization error is exactly the one that maximizes alignment. Finally, we show that task performance is an insufficient proxy for representational informativeness: a network with lower test error can retain less of the local geometry of its input. These findings reveal a non-trivial dependence of representational alignment on the quality and quantity of the training data and on the regularization of the learning problem.