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.

翻译：本文提出一个理论框架，将流形学习中的多图册自编码器与经典的向量丛及示性类理论联系起来。不同于将自编码器视为产生单一全局欧几里得嵌入的传统观点，我们将一组局部训练的编码器-解码器对视为流形上习得的图册。我们证明，任何重构一致的自编码器图册都能典范地定义满足上循环条件的转移映射，且当潜在维度与流形本征维度匹配时，这些转移映射的线性化将产生一个与切丛重合的向量丛。该构造为直接获取数据的微分拓扑不变量提供了途径。特别地，我们证明第一施蒂费尔-惠特尼类可通过习得转移映射的雅可比矩阵符号进行计算，从而为检测可定向性提供算法判据。我们还证明非平凡示性类对单图册表示构成阻碍，且自编码器图册的最小数量由流形的良覆盖结构决定。最后，我们将该方法应用于低维可定向与不可定向流形，以及一个高维不可定向图像数据集。