Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases
This work provides a novel theoretical framework for understanding the topological properties of data for researchers in manifold learning and topological data analysis, offering an algorithmic criterion for detecting orientability.
This paper establishes a connection between multi-chart autoencoders and the theory of vector bundles, showing that a collection of locally trained autoencoders can form a learned atlas. This framework allows for the computation of differential-topological invariants, such as the first Stiefel-Whitney class, which can detect orientability from the signs of Jacobian determinants of learned transition maps.
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.