Normalizing Flows on Quotient Manifolds via Boundary Quotients

Abstract

We introduce boundary quotients and present a framework for learning densities on manifolds that arise as boundary quotients of simpler domains. We show that this framework can be used to construct normalizing flows on quotient manifolds $N/G$, where a discrete group $G$ acts on $N$. We instantiate this construction for genus-$g$ surfaces $Σ_g$. When $G$ is finite, we show applicability to symmetry aware learning; we demonstrate this on cyclic quotients of the 3-sphere. Experiments on lens spaces show that simple pre-quotient RealNVP models can achieve strong results while being substantially cheaper to evaluate.