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Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases
🤖AI Summary
Researchers introduce a theoretical framework connecting multi-chart autoencoders in manifold learning with classical vector bundle theory and characteristic classes. The approach treats collections of locally trained encoder-decoder pairs as learned atlases on manifolds, enabling computation of differential-topological invariants and providing algorithmic criteria for detecting orientability.
Key Takeaways
- →Multi-chart autoencoders can be viewed as learned atlases on manifolds rather than single global Euclidean embeddings.
- →Reconstruction-consistent autoencoder atlases canonically define transition maps that satisfy the cocycle condition.
- →The first Stiefel-Whitney class can be computed from signs of Jacobians of learned transition maps to detect orientability.
- →Non-trivial characteristic classes provide obstructions to single-chart representations in autoencoder architectures.
- →The methodology successfully applies to both low-dimensional manifolds and high-dimensional non-orientable image datasets.
#autoencoder#manifold-learning#vector-bundles#machine-learning#topology#differential-geometry#arxiv#research
Read Original →via arXiv – CS AI
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