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  "path": "/abs/2602.22873v1",
  "publishedAt": "2026-02-27T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Eduardo Paluzo-Hidalgo",
    "Yuichi Ike"
  ],
  "textContent": "**Authors:** Eduardo Paluzo-Hidalgo, Yuichi Ike\n\nWe 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.",
  "title": "Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases"
}