  Deep generative models like variational autoencoders approximate the
intrinsic geometry of high dimensional data manifolds by learning
low-dimensional latent-space variables and an embedding function. The geometric
properties of these latent spaces has been studied under the lens of Riemannian
geometry; via analysis of the non-linearity of the generator function. In new
developments, deep generative models have been used for learning semantically
meaningful `disentangled' representations; that capture task relevant
attributes while being invariant to other attributes. In this work, we explore
the geometry of popular generative models for disentangled representation
learning. We use several metrics to compare the properties of latent spaces of
disentangled representation models in terms of class separability and curvature
of the latent-space. The results we obtain establish that the class
distinguishable features in the disentangled latent space exhibits higher
curvature as opposed to a variational autoencoder. We evaluate and compare the
geometry of three such models with variational autoencoder on two different
datasets. Further, our results show that distances and interpolation in the
latent space are significantly improved with Riemannian metrics derived from
the curvature of the space. We expect these results will have implications on
understanding how deep-networks can be made more robust, generalizable, as well
as interpretable.
