# Posterior Collapse

The work formulates dimension-wise posterior collapse in variational autoencoders as a fluctuation theory around partially collapsed states: treating the negative ELBO as an effective free energy, its quadratic expansion gives a Hessian acting as a mass matrix, collapsed directions form an invariant fluctuation sector whose exact mass spectrum follows from a conditional residual operator, and a local reactivation direction lowers the free energy once the decoder variance falls below the residual spectral upper edge, with equality marking marginality; numerical continuation shows latent dimensions lost near these spectral marginalities.

The work formulates dimension-wise posterior collapse in variational autoencoders as a fluctuation theory around partially collapsed states: treating the negative ELBO as an effective free energy, its quadratic expansion gives a Hessian acting as a mass matrix, collapsed directions form an invariant fluctuation sector whose exact mass spectrum follows from a conditional residual operator, and a local reactivation direction lowers the free energy once the decoder variance falls below the residual spectral upper edge, with equality marking marginality; numerical continuation shows latent dimensions lost near these spectral marginalities.