Incorporating geometric inductive biases into models can aid interpretability and generalization, but encoding to a specific geometric structure can be challenging due to the imposed topological constraints. In this paper, we theoretically and empirically characterize obstructions to training encoders with geometric latent spaces. We show that local optima can arise due to singularities (e.g. self-intersection) or due to an incorrect degree or winding number. We then discuss how normalizing flows can potentially circumvent these obstructions by defining multimodal variational distributions. Inspired by this observation, we propose a new flow-based model that maps data points to multimodal distributions over geometric spaces and empirically evaluate our model on 2 domains. We observe improved stability during training and a higher chance of converging to a homeomorphic encoder.

Deep latent-variable models learn representations of high-dimensional data in an unsupervised manner. A number of recent efforts have focused on learning representations that disentangle statistically independent axes of variation by introducing modifications to the standard objective function. These approaches generally assume a simple diagonal Gaussian prior and as a result are not able to reliably disentangle discrete factors of variation. We propose a two-level hierarchical objective to control relative degree of statistical independence between blocks of variables and individual variables within blocks. We derive this objective as a generalization of the evidence lower bound, which allows us to explicitly represent the trade-offs between mutual information between data and representation, KL divergence between representation and prior, and coverage of the support of the empirical data distribution. Experiments on a variety of datasets demonstrate that our objective can not only disentangle discrete variables, but that doing so also improves disentanglement of other variables and, importantly, generalization even to unseen combinations of factors.