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| "The spiked cumulant model introduces two latent factors, one recoverable by PCA from the covariance and one appearing only in higher-order moments and invisible to PCA (Section 3).", | |
| "For correlation exponent 2 ≤ k* < ∞, a minimal nonlinear autoencoder x̂ = (w/√d) σ(wᵀx/√d) with tied encoder-decoder weights provably recovers both latent spikes while PCA and linear autoencoders recover only the covariance-visible one (Section 2, Result 3.1).", | |
| "Approximate Message Passing weakly recovers both spikes once the sample-to-dimension ratio α exceeds a threshold α_weak^AMP determined by the moments of the coupling coefficients (Result 3.2).", | |
| "Under conditions C2 < 0 and C3 ≠ 0, spherical gradient flow on the nonlinear autoencoder achieves weak recovery of the hidden spike in logarithmic time Θ(log d) (Theorem 4.2).", | |
| "Linear autoencoders achieve lower reconstruction (test) loss than nonlinear autoencoders yet fail to recover the hidden spike, demonstrating that self-supervised test loss misaligns with representation quality (Figure 2)." | |
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