A solvable high-dimensional model where nonlinear autoencoders learn structure invisible to PCA while test loss misaligns with generalization

• URL: http://arxiv.org/abs/2602.10680v1
• Date: Wed, 11 Feb 2026 09:31:29 GMT
• Title: A solvable high-dimensional model where nonlinear autoencoders learn structure invisible to PCA while test loss misaligns with generalization
• Authors: Vicente Conde Mendes, Lorenzo Bardone, Cédric Koller, Jorge Medina Moreira, Vittorio Erba, Emanuele Troiani, Lenka Zdeborová,
• Abstract summary: We introduce a tractable high-dimensional spiked model with two latent factors.<n> PCA and linear autoencoders fail to recover the latter, while a minimal nonlinear autoencoder provably extracts both.<n>Our model also provides a tractable example where self-supervised test loss is poorly aligned with representation quality.
• Score: 12.791897914144378
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Many real-world datasets contain hidden structure that cannot be detected by simple linear correlations between input features. For example, latent factors may influence the data in a coordinated way, even though their effect is invisible to covariance-based methods such as PCA. In practice, nonlinear neural networks often succeed in extracting such hidden structure in unsupervised and self-supervised learning. However, constructing a minimal high-dimensional model where this advantage can be rigorously analyzed has remained an open theoretical challenge. We introduce a tractable high-dimensional spiked model with two latent factors: one visible to covariance, and one statistically dependent yet uncorrelated, appearing only in higher-order moments. PCA and linear autoencoders fail to recover the latter, while a minimal nonlinear autoencoder provably extracts both. We analyze both the population risk, and empirical risk minimization. Our model also provides a tractable example where self-supervised test loss is poorly aligned with representation quality: nonlinear autoencoders recover latent structure that linear methods miss, even though their reconstruction loss is higher.

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