This paper presents the geometric aspect of the autoencoder framework, which, despite its importance, has been relatively less recognized. Given a set of high-dimensional data points that approximately lie on some lower-dimensional manifold, an autoencoder learns the \textit{manifold} and its \textit{coordinate chart}, simultaneously. This geometric perspective naturally raises inquiries like "Does a finite set of data points correspond to a single manifold?" or "Is there only one coordinate chart that can represent the manifold?". The responses to these questions are negative, implying that there are multiple solution autoencoders given a dataset. Consequently, they sometimes produce incorrect manifolds with severely distorted latent space representations. In this paper, we introduce recent geometric approaches that address these issues.

翻译：本文阐述了自编码器框架的几何特性，尽管该特性至关重要，但至今仍未被充分认知。给定一组近似位于低维流形上的高维数据点，自编码器能够同时学习该流形及其坐标图。这种几何视角自然引发了诸如"有限数据点集合是否对应单一流形？"或"是否存在唯一能表征该流形的坐标图？"等疑问。对这些问题的回答是否定的——这意味着给定数据集存在多个解自编码器。因此，它们有时会产生具有严重畸变潜在空间表征的错误流形。本文介绍了近年来针对这些问题提出的几何方法。