Learning geometry and topology via multi-chart flows

2025-05-30

AI Summary

Standard normalizing flows fail to model low-dimensional Riemannian manifolds embedded in high-dimensional spaces with nontrivial topology (e.g., tori, Klein bottles), due to their inherent topological triviality. Method: We propose a multi-atlas normalizing flow framework that overcomes this limitation by decomposing the manifold into overlapping coordinate charts. Our approach introduces a novel collaborative training paradigm for local flows, enforces consistency constraints on overlapping regions, defines local coordinate mappings, and incorporates a custom geodesic numerical integration algorithm tailored to manifold geometry—integrated with variational inference for joint geometric-topological modeling. Contribution/Results: Experiments demonstrate that our framework accurately recovers both homotopy and homology structures of complex manifolds. It significantly outperforms existing methods on topological estimation tasks, establishing a principled foundation for learning distributions on topologically nontrivial manifolds.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Other Foundations of Computer Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Learning low-dimensional manifolds with non-trivial topology
• Developing multi-chart flows for manifold representation
• Computing geodesics on learned topological manifolds

Innovation

• Multi-chart flows for manifold learning
• General training scheme for multiple flows
• Numerical algorithms for geodesic computation