Don't Blame the ELBO! A Linear VAE Perspective on Posterior Collapse
Abstract
Posterior collapse in Variational Autoencoders (VAEs) arises when the variational posterior distribution closely matches the prior for a subset of latent variables. This paper presents a simple and intuitive explanation for posterior collapse through the analysis of linear VAEs and their direct correspondence with Probabilistic PCA (pPCA). We explain how posterior collapse may occur in pPCA due to local maxima in the log marginal likelihood. Unexpectedly, we prove that the ELBO objective for the linear VAE does not introduce additional spurious local maxima relative to log marginal likelihood. We show further that training a linear VAE with exact variational inference recovers an identifiable global maximum corresponding to the principal component directions. Empirically, we find that our linear analysis is predictive even for highcapacity, nonlinear VAEs and helps explain the relationship between the observation noise, local maxima, and posterior collapse in deep Gaussian VAEs.
 Publication:

arXiv eprints
 Pub Date:
 November 2019
 arXiv:
 arXiv:1911.02469
 Bibcode:
 2019arXiv191102469L
 Keywords:

 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 11 main pages, 10 appendix pages. 13 figures total. Accepted at 33rd Conference on Neural Information Processing Systems (NeurIPS 2019)