Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases

• URL: http://arxiv.org/abs/2602.22873v1
• Date: Thu, 26 Feb 2026 11:10:35 GMT
• Title: Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases
• Authors: Eduardo Paluzo-Hidalgo, Yuichi Ike,
• Abstract summary: We introduce a theoretical framework that connects multi-chart autoencoders in manifold learning with the classical theory of vector bundles and characteristic classes.<n>We show that any reconstruction-consistent autoencoder atlas canonically defines transition maps satisfying the cocycle condition.<n>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.
• Score: 4.265353613980049
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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