使用自编码器图谱学习切丛和示性类
Learning Tangent Bundles and Characteristic Classes with Autoencoder Atlases
AI总结：
本文提出一个理论框架，将流形学习中的多图自编码器与向量丛和示性类的经典理论联系起来，通过自编码器图谱定义转移映射并计算第一Stiefel-Whitney类，从而检测数据可定向性。
摘要：
我们引入了一个理论框架，将流形学习中的多图自编码器与向量丛和示性类的经典理论联系起来。我们不将自编码器视为产生单个全局欧几里得嵌入，而是将一组局部训练的编码器-解码器对视为流形上的学习图谱。我们证明，任何重建一致的自编码器图谱都能典范地定义满足上循环条件的转移映射，并且将这些转移映射线性化会得到一个向量丛，当潜在维度与流形的内在维度匹配时，该向量丛与切丛一致。这种构造提供了对数据的微分拓扑不变量的直接访问。特别地，我们证明第一Stiefel-Whitney类可以从学习到的转移映射的雅可比行列式的符号计算出来，从而得到检测可定向性的算法准则。我们还证明，非平凡的示性类对单图表示构成障碍，并且自编码器图的最小数量由流形的良好覆盖结构决定。最后，我们将我们的方法应用于低维可定向和不可定向流形，以及一个不可定向的高维图像数据集。
英文摘要：
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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