When Nonlinear Autoencoders Learn Structure Invisible to PCA

Independent headline

0.119 vs 0.019

hidden-spike overlap for a tied ReLU autoencoder versus PCA at d=256.

The nonlinear representation sees a higher-order latent dependence that covariance cannot, despite paying a higher reconstruction loss.

alpha = 64 seedstied one-neuron AE

What was tested

Spiked-cumulant data at d=64, 128, and 256.
ReLU ERM, PCA baselines, and held-out reconstruction losses.
Linearized AMP state evolution and the leading population-flow ODE.

Evidence

Visible and hidden spike recovery — PCA retains the covariance-visible spike while nonlinear training develops nonzero hidden-spike overlap.

Claim audit

Verified

PCA invisibility and the test-loss mismatch persist from d=64 through d=256.

Scoped or contradicted

Full finite-dimensional AMP was not rerun. The extracted claim is broader than the theorem: hidden-spike recovery requires activation-specific moment conditions.

Reproducibility boundary

The CPU run reaches d=256, not d=2,000; it verifies weak recovery but not the asymptotic overlap curve.