File size: 14,979 Bytes
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"destructive_or_boundary_control": "Removing only the S whitening makes the v direction covariance-visible and forces the top-PC hidden overlap above 0.75.",
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"independent_oracle": "Parity gives E[lambda nu]=0 independently, while quadrature evaluates E[lambda^2 nu]; direct top-PC overlaps separately measure the covariance-visible and hidden directions.",
"limitation": "The finite-d empirical covariance test instantiates the paper's synthetic model at d=256; it does not claim a new real-data cumulant benchmark.",
"literal_claim": "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).",
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"result": "E[lambda nu]=0.000e+00, E[lambda^2 nu]=0.817310; S-whitened PCA overlaps u=0.8458, v=0.0263; unwhitened v control=0.9747.",
"scope_boundary": "The finite-d empirical covariance test instantiates the paper's synthetic model at d=256; it does not claim a new real-data cumulant benchmark.",
"source_locator": "arXiv 2602.10680 and SPOC-group/advantage_nonlinearity commit 437801797152a9a70ea5839cad382ed68515915e"
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"limitation": "This is the released finite-dimensional k*=2 instance and one activation from the theorem's successful class; it is not an empirical proof for every finite exponent or activation.",
"literal_claim": "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).",
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"result": "At alpha=8 the nonlinear model overlaps u=0.8023, v=0.1865, versus PCA u=0.8826, v=0.0250; minimum alpha>=6 hidden advantage is 0.0518.",
"scope_boundary": "This is the released finite-dimensional k*=2 instance and one activation from the theorem's successful class; it is not an empirical proof for every finite exponent or activation.",
"source_locator": "arXiv 2602.10680 and SPOC-group/advantage_nonlinearity commit 437801797152a9a70ea5839cad382ed68515915e"
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{
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"destructive_or_boundary_control": "Setting delta=0 makes the two latents independent and removes every finite mixed moment while leaving dimension, alpha, AMP code, and signal strengths unchanged.",
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"limitation": "The executed threshold sweep is d=256 with six seeds, not the paper's d=10,000/72-seed figure; it executes the same released AMP and channel rather than a proxy algorithm.",
"literal_claim": "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).",
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"result": "Mean AMP recovery grows from alpha=1 (u=0.0717, v=0.0663) to alpha=5 (u=0.8151, v=0.4117); the independent-latent hidden control is 0.0543.",
"scope_boundary": "The executed threshold sweep is d=256 with six seeds, not the paper's d=10,000/72-seed figure; it executes the same released AMP and channel rather than a proxy algorithm.",
"source_locator": "arXiv 2602.10680 and SPOC-group/advantage_nonlinearity commit 437801797152a9a70ea5839cad382ed68515915e"
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"destructive_or_boundary_control": "Replacing nu(lambda) with an independent Rademacher latent keeps C2 unchanged but drives C3 to numerical zero; sqrt(d) is also fitted as a wrong-rate control.",
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"independent_oracle": "A fourth-order local polynomial independently recovers the theorem coefficients, while direct weak-recovery hitting times are compared under log(d) and sqrt(d) regressors.",
"limitation": "The overlap ODE follows the theorem's early-stage fourth-order population expansion up to weak recovery 0.02; it does not extrapolate beyond that local regime or substitute for empirical ERM.",
"literal_claim": "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).",
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"result": "C2=-0.159824, C3=-0.010910, independent C3=-5.259e-15; hitting-time log(d) R2=0.994504 versus sqrt(d) R2=0.771150.",
"scope_boundary": "The overlap ODE follows the theorem's early-stage fourth-order population expansion up to weak recovery 0.02; it does not extrapolate beyond that local regime or substitute for empirical ERM.",
"source_locator": "arXiv 2602.10680 and SPOC-group/advantage_nonlinearity commit 437801797152a9a70ea5839cad382ed68515915e"
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"independent_oracle": "Held-out squared reconstruction loss is computed by the authors' reconstruction_loss function, while planted hidden-spike cosine is evaluated independently from the learned weights.",
"limitation": "The reversal is measured on the paper's controlled spiked-cumulant task at alpha=6; it does not imply reconstruction loss is universally misaligned in all self-supervised systems.",
"literal_claim": "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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"result": "Every seed has lower linear loss and higher nonlinear hidden recovery: mean loss linear=127.6445 versus nonlinear=128.1300; mean hidden overlap linear=0.0263 versus nonlinear=0.0782.",
"scope_boundary": "The reversal is measured on the paper's controlled spiked-cumulant task at alpha=6; it does not imply reconstruction loss is universally misaligned in all self-supervised systems.",
"source_locator": "arXiv 2602.10680 and SPOC-group/advantage_nonlinearity commit 437801797152a9a70ea5839cad382ed68515915e"
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"paper_id": "wm3ABfhE7P",
"paper_title": "A Solvable High-Dimensional Model Where Nonlinear Autoencoders Learn Structure Invisible to PCA While Test Loss Misaligns With Generalization",
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