Learning Geometry and Topology via Multi-Chart Flows

TL;DR

This paper introduces a method for learning multiple normalizing flows to model data on manifolds with complex topology and provides algorithms for geodesic computation, improving topology estimation.

Contribution

It proposes a general training scheme for multi-chart flows and develops algorithms for geodesic computation on manifolds with non-trivial topology.

Findings

Significant improvements in topology estimation using multi-chart flows.

First numerical algorithms for geodesic computation on such manifolds.

Enhanced modeling of data on complex topological manifolds.

Abstract

Real world data often lie on low-dimensional Riemannian manifolds embedded in high-dimensional spaces. This motivates learning degenerate normalizing flows that map between the ambient space and a low-dimensional latent space. However, if the manifold has a non-trivial topology, it can never be correctly learned using a single flow. Instead multiple flows must be `glued together'. In this paper, we first propose the general training scheme for learning such a collection of flows, and secondly we develop the first numerical algorithms for computing geodesics on such manifolds. Empirically, we demonstrate that this leads to highly significant improvements in topology estimation.