Covering-Space Normalizing Flows: Approximating Pushforwards on Lens Spaces

TL;DR

This paper introduces a method for constructing and approximating pushforward distributions on lens spaces using normalizing flows on S^3, effectively reducing redundancies in symmetric cases.

Contribution

The authors develop a novel approach leveraging the universal covering map to approximate distributions on lens spaces with flows on S^3, addressing symmetry redundancies.

Findings

Successfully approximated von Mises-Fisher pushforwards.

Modeled a Z_12-symmetric Boltzmann distribution for benzene.

Reduced redundancies in symmetric S^3 distributions.

Abstract

We construct pushforward distributions via the universal covering map rho: S^3 -> L(p;q) with the goal of approximating these distributions using flows on L(p;q). We highlight that our method deletes redundancies in the case of a symmetric S^3 distribution. Using our model, we approximate the pushforwards of von Mises-Fisher-induced target densities as well as that of a Z_12-symmetric Boltzmann distribution on S^3 constructed to model benzene.