  Boltzmann machines are powerful distributions that have been shown to be an
effective prior over binary latent variables in variational autoencoders
(VAEs). However, previous methods for training discrete VAEs have used the
evidence lower bound and not the tighter importance-weighted bound. We propose
two approaches for relaxing Boltzmann machines to continuous distributions that
permit training with importance-weighted bounds. These relaxations are based on
generalized overlapping transformations and the Gaussian integral trick.
Experiments on the MNIST and OMNIGLOT datasets show that these relaxations
outperform previous discrete VAEs with Boltzmann priors. An implementation
which reproduces these results is available at
https://github.com/QuadrantAI/dvae .
