Learning geometry and topology via multi-chart flows
This addresses a fundamental challenge in manifold learning for data science, enabling more accurate modeling of complex real-world data structures.
The paper tackles the problem of learning low-dimensional Riemannian manifolds with non-trivial topology from high-dimensional data by proposing a multi-chart flow training scheme and numerical algorithms for geodesic computation, resulting in highly significant improvements in topology estimation.
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.