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This paper provides a rigorous theoretical framework for understanding dimension-wise posterior collapse in variational autoencoders by modeling it as a fluctuation theory around partially collapsed states. By deriving the exact mass spectrum of latent fluctuations, the authors establish a spectral criterion that predicts when and how latent dimensions will successively lose stability. These findings offer deep mathematical insights into the mechanics of representation learning, specifically addressing the physical principles governing model collapse in generative AI systems.
Residual spectral instabilities in representation learning
arXiv:2610.11257v1 Announce Type: new Abstract: Learned representations can lose latent degrees of freedom successively, suggesting a cascade of transitions whose underlying stability principle remains unclear. Here we formulate dimension-wise posterior collapse in variational autoencoder (VAE) as a fluctuation theory around partially collapsed states. Interpreting the negative evidence lower bound as an effective free energy, its quadratic expansion defines a Gaussian theory whose Hessian acts as a mass matrix for latent fluctuations. We show that the collapsed directions form an invariant fluctuation sector and derive its exact mass spectrum in terms of a conditional residual operator. A local reactivation direction lowers the free energy when the decoder variance falls below the residual spectral upper edge, with equality marking marginality. The criterion recovers principal component thresholds in the linear Gaussian VAE limit. Viewed in reverse along continuously connected branches, the reactivation boundary provides a local criterion for successive collapse. Numerical continuation experiments show successive loss of latent dimensions near these spectral marginalities. These results support a spectral cascade interpretation governed by residual information left unexplained by the surviving representation.
Abstract:Learned representations can lose latent degrees of freedom successively, suggesting a cascade of transitions whose underlying stability principle remains unclear. Here we formulate dimension-wise posterior collapse in variational autoencoder (VAE) as a fluctuation theory around partially collapsed states. Interpreting the negative evidence lower bound as an effective free energy, its quadratic expansion defines a Gaussian theory whose Hessian acts as a mass matrix for latent fluctuations. We show that the collapsed directions form an invariant fluctuation sector and derive its exact mass spectrum in terms of a conditional residual operator. A local reactivation direction lowers the free energy when the decoder variance falls below the residual spectral upper edge, with equality marking marginality. The criterion recovers principal component thresholds in the linear Gaussian VAE limit. Viewed in reverse along continuously connected branches, the reactivation boundary provides a local criterion for successive collapse. Numerical continuation experiments show successive loss of latent dimensions near these spectral marginalities. These results support a spectral cascade interpretation governed by residual information left unexplained by the surviving representation.
Machine Learning (cs.LG); Statistical Mechanics (cond-mat.stat-mech)
arXiv:2610.11257 [cs.LG]
(or arXiv:2610.11257v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.11257 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 269 words
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