  We assume that a high-dimensional datum, like an image, is a compositional
expression of a set of properties, with a complicated non-linear relationship
between the datum and its properties. This paper proposes a factorial mixture
prior for capturing latent properties, thereby adding structured
compositionality to deep generative models. The prior treats a latent vector as
belonging to Cartesian product of subspaces, each of which is quantized
separately with a Gaussian mixture model. Some mixture components can be set to
represent properties as observed random variables whenever labeled properties
are present. Through a combination of stochastic variational inference and
gradient descent, a method for learning how to infer discrete properties in an
unsupervised or semi-supervised way is outlined and empirically evaluated.
