Continuously Parameterized Mixture Models

Mixture models are universal approximators of smooth densities but are difficult to utilize in complicated datasets due to restrictions on typically available modes and challenges with initialiations. We show that by continuously parameterizing a mixture of factor analyzers using a learned ordinary differential equation, we can improve the fit of mixture models over direct methods. Once trained, the mixture components can be extracted and the neural ODE can be discarded, leaving us with an effective, but low-resource model. We additionally explore the use of a training curriculum from an easy-to-model latent space extracted from a normalizing flow to the more complex input space and show that the smooth curriculum helps to stabilize and improve results with and without the continuous parameterization. Finally, we introduce a hierarchical version of the model to enable more flexible, robust classification and clustering, and show substantial improvements against traditional parameterizations of GMMs.