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 $\displaystyle n^{-\frac{2}{d+2}}\log^4 n$, 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.

翻译：自编码器在各种应用中对高维数据学习低维潜在特征方面展现了显著成功。假设数据采样自一个低维流形附近，我们采用图表自编码器，将数据编码到一组图表上的低维潜在特征中，从而保持数据流形的拓扑和几何结构。本文建立了图表自编码器泛化误差的统计保证，并通过考虑 $d$ 维流形上的 $n$ 个含噪训练样本及其无噪对应样本，展示了其去噪能力。通过训练自编码器，我们证明图表自编码器能有效去除含正态噪声的输入数据。在合适的网络架构下，我们证明图表自编码器实现了 $\displaystyle n^{-\frac{2}{d+2}}\log^4 n$ 量级的平方泛化误差，该误差取决于流形的内蕴维度，并仅弱依赖于环境维度与噪声水平。我们进一步将理论推广至同时包含法向和切向分量的噪声数据，此时图表自编码器对法向分量仍表现出去噪效果。作为特例，只要数据流形具有全局参数化，我们的理论也适用于经典自编码器。这些结果为自编码器的有效性奠定了坚实的理论基础，并通过多项数值实验得到进一步验证。