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

TL;DR

This paper introduces a high-dimensional model demonstrating how nonlinear autoencoders can uncover hidden data structures invisible to PCA, revealing a misalignment between test loss and representation quality.

Contribution

The paper provides a rigorous theoretical analysis of a high-dimensional model showing nonlinear autoencoders' ability to learn complex structures beyond PCA's reach.

Findings

Nonlinear autoencoders recover hidden latent factors missed by PCA.

Test loss does not always correlate with the quality of learned representations.

Linear methods fail to detect certain higher-order dependencies in data.

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.