Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases

2026-02-26

AI Summary

This work proposes a multi-chart autoencoder framework to learn the differential topological structure of data manifolds, with a focus on their tangent bundles and Stiefel–Whitney characteristic classes. By constructing local coordinate charts that satisfy cocycle conditions together with associated transition maps, and leveraging the signs of the Jacobian matrices of these transitions, the study establishes—for the first time—a rigorous connection between autoencoder atlases and vector bundle theory. This enables an algorithmic computation of the first Stiefel–Whitney class, thereby determining manifold orientability and revealing how characteristic classes obstruct global single-chart representations. Experiments on both low- and high-dimensional non-orientable manifolds—including image datasets—demonstrate the method’s effectiveness in accurately detecting orientability and identifying the minimal atlas size required for faithful representation.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes. Rather than viewing autoencoders as producing a single global Euclidean embedding, we treat a collection of locally trained encoder-decoder pairs as a learned atlas on a manifold. We show that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition, and that linearising these transition maps yields a vector bundle coinciding with the tangent bundle when the latent dimension matches the intrinsic dimension of the manifold. This construction provides direct access to differential-topological invariants of the data. In particular, we show that the first Stiefel-Whitney class can be computed from the signs of the Jacobians of learned transition maps, yielding an algorithmic criterion for detecting orientability. We also show that non-trivial characteristic classes provide obstructions to single-chart representations, and that the minimum number of autoencoder charts is determined by the good cover structure of the manifold. Finally, we apply our methodology to low-dimensional orientable and non-orientable manifolds, as well as to a non-orientable high-dimensional image dataset.

Problem

• tangent bundle
• characteristic classes
• manifold learning
• orientability
• autoencoder atlas

Innovation

• autoencoder atlas
• tangent bundle
• characteristic classes
• Stiefel-Whitney class
• manifold learning