Geometry of Deep Generative Models for Disentangled Representations

Deep generative models like variational autoencoders approximate the intrinsic geometry of high dimensional data manifolds by learning a set of low-dimensional latent-space variables and an embedding function. The geometrical 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 proposed study establishes 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. The proposed study shows that the distances and interpolations in the latent space are significantly improved with Riemannian metric owing to the curvature of the space.