  Deep generative models are tremendously successful in learning
low-dimensional latent representations that well-describe the data. These
representations, however, tend to much distort relationships between points,
i.e. pairwise distances tend to not reflect semantic similarities well. This
renders unsupervised tasks, such as clustering, difficult when working with the
latent representations. We demonstrate that taking the geometry of the
generative model into account is sufficient to make simple clustering
algorithms work well over latent representations. Leaning on the recent finding
that deep generative models constitute stochastically immersed Riemannian
manifolds, we propose an efficient algorithm for computing geodesics (shortest
paths) and computing distances in the latent space, while taking its distortion
into account. We further propose a new architecture for modeling uncertainty in
variational autoencoders, which is essential for understanding the geometry of
deep generative models. Experiments show that the geodesic distance is very
likely to reflect the internal structure of the data.
