Autoencoders have demonstrated remarkable success in learning low-dimensional latent features of high-dimensional data across various applications. Assuming that data are sampled near a low-dimensional manifold, we employ chart autoencoders, which encode data into low-dimensional latent features on a collection of charts, preserving the topology and geometry of the data manifold. Our paper establishes statistical guarantees on the generalization error of chart autoencoders, and we demonstrate their denoising capabilities by considering n noisy training samples, along with their noise-free counterparts, on a d-dimensional manifold. By training autoencoders, we show that chart autoencoders can effectively denoise the input data with normal noise. We prove that, under proper network architectures, chart autoencoders achieve a squared generalization error in the order of $n^{-\frac{2}{d+2}}{\log }^{4}n$, which depends on the intrinsic dimension of the manifold and only weakly depends on the ambient dimension and noise level. We further extend our theory on data with noise containing both normal and tangential components, where chart autoencoders still exhibit a denoising effect for the normal component. As a special case, our theory also applies to classical autoencoders, as long as the data manifold has a global parametrization. Our results provide a solid theoretical foundation for the effectiveness of autoencoders, which is further validated through several numerical experiments.

自动编码器在跨各种应用程序学习高维数据的低维潜在特征方面取得了显着的成功。假设数据在低维流形附近采样，我们采用图表自动编码器，将数据编码为图表集合上的低维潜在特征，保留数据流形的拓扑和几何形状。我们的论文对图表自动编码器的泛化误差建立了统计保证，并且通过在d维流形上考虑n 个噪声训练样本及其无噪声对应样本，展示了它们的去噪能力。通过训练自动编码器，我们表明图表自动编码器可以有效地用正常噪声对输入数据进行去噪。我们证明，在适当的网络架构下，图表自动编码器可以实现平方泛化误差$n^{-\frac{2}{d+2}}{\mathrm{日志}}^{4}n$，这取决于流形的固有尺寸，并且仅微弱地取决于环境尺寸和噪声水平。我们进一步扩展了我们关于包含法向分量和切向分量的噪声的数据的理论，其中图表自动编码器仍然对法向分量表现出去噪效果。作为一个特例，我们的理论也适用于经典自动编码器，只要数据流形具有全局参数化。我们的结果为自动编码器的有效性提供了坚实的理论基础，并通过多个数值实验进一步验证。