Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases

TL;DR

This paper develops a theoretical framework linking multi-chart autoencoders in manifold learning with vector bundles and characteristic classes, enabling topological analysis of data manifolds through learned transition maps.

Contribution

It introduces a novel perspective of autoencoder collections as learned atlases, connecting them to tangent bundles and topological invariants, and provides algorithms for detecting orientability and characteristic classes.

Findings

Transition maps satisfy cocycle conditions.

First Stiefel-Whitney class can be computed from Jacobian signs.

Non-trivial characteristic classes obstruct single-chart representations.

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.