  Recent work in unsupervised learning has focused on efficient inference and
learning in latent variables models. Training these models by maximizing the
evidence (marginal likelihood) is typically intractable. Thus, a common
approximation is to maximize the Evidence Lower BOund (ELBO) instead.
Variational autoencoders (VAE) are a powerful and widely-used class of
generative models that optimize the ELBO efficiently for large datasets.
However, the VAE's default Gaussian choice for the prior imposes a strong
constraint on its ability to represent the true posterior, thereby degrading
overall performance. A Gaussian mixture model (GMM) would be a richer prior,
but cannot be handled efficiently within the VAE framework because of the
intractability of the Kullback-Leibler divergence for GMMs. We deviate from the
common VAE framework in favor of one with an analytical solution for Gaussian
mixture prior. To perform efficient inference for GMM priors, we introduce a
new constrained objective based on the Cauchy-Schwarz divergence, which can be
computed analytically for GMMs. This new objective allows us to incorporate
richer, multi-modal priors into the autoencoding framework. We provide
empirical studies on a range of datasets and show that our objective improves
upon variational auto-encoding models in density estimation, unsupervised
clustering, semi-supervised learning, and face analysis.
