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The posterior collapse phenomenon in variational autoencoder (VAE), where thevariational posterior distribution closely matches the prior distribution, canhinder the quality of the learned latent variables. As a consequence ofposterior collapse, the latent variables extracted by the encoder in VAEpreserve less information from the input data and thus fail to producemeaningful representations as input to the reconstruction process in thedecoder. While this phenomenon has been an actively addressed topic related toVAE performance, the theory for posterior collapse remains underdeveloped,especially beyond the standard VAE. In this work, we advance the theoreticalunderstanding of posterior collapse to two important and prevalent yet lessstudied classes of VAE: conditional VAE and hierarchical VAE. Specifically, viaa non-trivial theoretical analysis of linear conditional VAE and hierarchicalVAE with two levels of latent, we prove that the cause of posterior collapsesin these models includes the correlation between the input and output of theconditional VAE and the effect of learnable encoder variance in thehierarchical VAE. We empirically validate our theoretical findings for linearconditional and hierarchical VAE and demonstrate that these results are alsopredictive for non-linear cases with extensive experiments.

Beyond Vanilla Variational Autoencoders: Detecting Posterior Collapse in Conditional and Hierarchical Variational Autoencoders