  We take steps towards understanding the "posterior collapse (PC)" difficulty
in variational autoencoders (VAEs),~i.e. a degenerate optimum in which the
latent codes become independent of their corresponding inputs. We rely on
calculus of variations and theoretically explore a few popular VAE models,
showing that PC always occurs for non-parametric encoders and decoders.
Inspired by the popular noise contrastive estimation algorithm, we propose
NC-VAE where the encoder discriminates between the latent codes of real data
and of some artificially generated noise, in addition to encouraging good data
reconstruction abilities. Theoretically, we prove that our model cannot reach
PC and provide novel lower bounds. Our method is straightforward to implement
and has the same run-time as vanilla VAE. Empirically, we showcase its benefits
on popular image and text datasets.
