Controlled approximations and inter-branch relations for residual spectra

Develop controlled approximations to the nonlinear stationary background states of Gaussian variational autoencoders and determine relations between the residual spectra associated with successive stationary branches.

Background

The residual spectral upper edge governing collapsed-sector stability depends on the stationary decoder and active variational posterior, which satisfy nonlinear self-consistency equations involving high-dimensional integrals. Outside special cases such as the linear Gaussian model, these equations generally cannot be reduced to expressions involving only simple statistics of the data.

The unresolved problem is therefore twofold: obtain controlled approximations for the stationary background state and characterize how residual spectra change between successive stationary branches. Such results would clarify the spectral organization of dimension-wise posterior collapse beyond the exactly solvable linear setting.