  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.
