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[2602.22873] Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases
Summary
This paper introduces a framework connecting multi-chart autoencoders with vector bundles and characteristic classes, enhancing manifold learning techniques.
Why It Matters
The research provides a novel approach to understanding manifold learning through the lens of algebraic topology, potentially improving algorithms for data representation and analysis in high-dimensional spaces. This could have significant implications for fields like computer vision and AI.
Key Takeaways
- Autoencoders can be viewed as learned atlases on manifolds, enhancing manifold learning.
- The framework allows for the computation of differential-topological invariants, aiding in data analysis.
- The study provides a method for detecting orientability using the first Stiefel-Whitney class.
- Non-trivial characteristic classes can reveal limitations of single-chart representations.
- The minimum number of autoencoder charts is linked to the manifold's good cover structure.
Mathematics > Algebraic Topology arXiv:2602.22873 (math) [Submitted on 26 Feb 2026] Title:Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases Authors:Eduardo Paluzo-Hidalgo, Yuichi Ike View a PDF of the paper titled Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases, by Eduardo Paluzo-Hidalgo and 1 other authors View PDF HTML (experimental) 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 autoenco...