  In this conceptual work, we present Deep Convolutional Gaussian Mixture
Models (DCGMMs): a new formulation of deep hierarchical Gaussian Mixture Models
(GMMs) that is particularly suitable for describing and generating images.
Vanilla (i.e., flat) GMMs require a very large number of components to describe
images well, leading to long training times and memory issues. DCGMMs avoid
this by a stacked architecture of multiple GMM layers, linked by convolution
and pooling operations. This allows to exploit the compositionality of images
in a similar way as deep CNNs do. DCGMMs can be trained end-to-end by
Stochastic Gradient Descent. This sets them apart from vanilla GMMs which are
trained by Expectation-Maximization, requiring a prior k-means initialization
which is infeasible in a layered structure. For generating sharp images with
DCGMMs, we introduce a new gradient-based technique for sampling through
non-invertible operations like convolution and pooling. Based on the MNIST and
FashionMNIST datasets, we validate the DCGMMs model by demonstrating its
superiority over flat GMMs for clustering, sampling and outlier detection.
