通过 Chart Autoencoders 进行内在数据结构的深度非参数估计：泛化误差与鲁棒性
Deep Nonparametric Estimation of Intrinsic Data Structures by Chart Autoencoders: Generalization Error and Robustness
AI总结：
本文提出采用 chart autoencoders 保留数据流形的拓扑与几何结构，证明了其在噪声数据下的泛化误差仅微弱依赖环境维度，并具备法向去噪能力，为 autoencoders 的有效性提供了理论基础与实验验证。
摘要：
Autoencoders 在各种应用中学习高维数据的低维潜在特征方面取得了显著成功。假设数据在低维流形附近采样，我们采用 chart autoencoders，它们将数据编码为图表集合上的低维潜在特征，从而保留数据流形的拓扑和几何结构。我们的论文建立了关于 chart autoencoders 泛化误差的统计保证，并通过考虑 $d$ 维流形上的 $n$ 个噪声训练样本及其无噪声对应样本来证明其去噪能力。通过训练 autoencoders，我们表明 chart autoencoders 能够有效去除输入数据中的法向噪声。我们证明，在适当的网络架构下，chart autoencoders 实现了数量级为 $\displaystyle n^{-\frac{2}{d+2}}\log^4 n$ 的平方泛化误差，该误差取决于流形的内在维度，且仅微弱地依赖于环境维度和噪声水平。我们进一步将理论扩展到包含法向和切向分量的噪声数据，其中 chart autoencoders 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，我们的理论同样适用于经典 autoencoders。我们的结果为 autoencoders 的有效性提供了坚实的理论基础，并通过若干数值实验进一步验证了这一点。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
