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Covering-Space Normalizing Flows: Approximating Pushforwards on Lens Spaces

William Ghanem

TL;DR

The paper develops a framework to learn distributions on lens spaces by pushing forward distributions from the universal cover $S^3$ through the covering map to $L(p;q)$, using a genus-1 Heegaard splitting to train separate flows on two solid tori. By symmetrizing the $S^3$ density to respect deck actions, the method eliminates redundant symmetries and preserves local smoothness across the gluing boundary, enabling accurate pushforwards with manageable topology. Three experiments–two mitiga­tions with von Mises–Fisher components and a benzene-related Boltzmann distribution–demonstrate small KL divergences and clear symmetry-deletion of modes, illustrating practical utility for symmetry-reduced configuration spaces. The approach provides a principled path to learn on quotient manifolds and can be extended to other covering-space scenarios and manifolds with Heegaard decompositions. The results have potential impact for molecular configuration modeling and other domains involving quotient geometries where topological complexity would otherwise hinder learning.

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.

Covering-Space Normalizing Flows: Approximating Pushforwards on Lens Spaces

TL;DR

The paper develops a framework to learn distributions on lens spaces by pushing forward distributions from the universal cover through the covering map to , using a genus-1 Heegaard splitting to train separate flows on two solid tori. By symmetrizing the density to respect deck actions, the method eliminates redundant symmetries and preserves local smoothness across the gluing boundary, enabling accurate pushforwards with manageable topology. Three experiments–two mitiga­tions with von Mises–Fisher components and a benzene-related Boltzmann distribution–demonstrate small KL divergences and clear symmetry-deletion of modes, illustrating practical utility for symmetry-reduced configuration spaces. The approach provides a principled path to learn on quotient manifolds and can be extended to other covering-space scenarios and manifolds with Heegaard decompositions. The results have potential impact for molecular configuration modeling and other domains involving quotient geometries where topological complexity would otherwise hinder learning.

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.

Paper Structure

This paper contains 15 sections, 2 theorems, 31 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Normalizing Flows
  4. Lens Spaces and Heegaard Splittings
  5. Method
  6. Pushforward from $S^3$
  7. Architecture and Training Methods
  8. Experiments
  9. Discussion
  10. Conclusion
  11. Appendix
  12. Construction of the Pushforward
  13. Training Architecture Details
  14. Post Training Metrics
  15. Potential Function

Key Result

Proposition 2

Given a finite sheeted covering map $\rho:M \rightarrow N$ and a distribution $\mu_N$ on $N$, there exists a distribution $\mu_M$ on $M$ such that $\mu_N=\rho_{*}\mu_M$.

Figures (1)

  • Figure 1: Samples from the the learned distribution on $T_1 \cup _AT_2$ associated to the pushforwards of $p_{S_3}^1$ (left), $p_{S_3}^2$ (middle), and $p_{S^{3}}^{Boltz}$ (right) visualized on each $T_i$. Only samples within the top 1% of pushforward pdf values are shown to highlight the high density regions. Higher density is in yellow, lower density is red.

Theorems & Definitions (5)

  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof