Normalizing Flows on Quotient Manifolds via Boundary Quotients

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.