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 high-capacity, non-linear VAEs and helps explain the relationship between the observation noise, local maxima, and posterior collapse in deep Gaussian VAEs.

arXiv comment11 main pages, 10 appendix pages. 13 figures total. Accepted at 33rd Conference on Neural Information Processing Systems (NeurIPS 2019)

References in corpus (13)

• Adam: A Method for Stochastic Optimization
• Stochastic Backpropagation and Approximate Inference in Deep Generative Models
• Lagging Inference Networks and Posterior Collapse in Variational Autoencoders
• BIVA: A Very Deep Hierarchy of Latent Variables for Generative Modeling
• Diagnosing and Enhancing VAE Models
• Inference Suboptimality in Variational Autoencoders
• VAE with a VampPrior
• Tackling Over-pruning in Variational Autoencoders
• Fixing a Broken ELBO
• Avoiding Latent Variable Collapse With Generative Skip Models
• Semi-Amortized Variational Autoencoders
• Loss Landscapes of Regularized Linear Autoencoders
• Hidden Talents of the Variational Autoencoder