  Discrete random variables are natural components of probabilistic clustering
models. A number of VAE variants with discrete latent variables have been
developed. Training such methods requires marginalizing over the discrete
latent variables, causing training time complexity to be linear in the number
clusters. By applying a continuous relaxation to the discrete variables in
these methods we can achieve a reduction in the training time complexity to be
constant in the number of clusters used. We demonstrate that in practice for
one such method, the Gaussian Mixture VAE, the use of a continuous relaxation
has no negative effect on the quality of the clustering but provides a
substantial reduction in training time, reducing training time on CIFAR-100
with 20 clusters from 47 hours to less than 6 hours.
