Autoencoding is a popular method in representation learning. Conventional autoencoders employ symmetric encoding-decoding procedures and a simple Euclidean latent space to detect hidden low-dimensional structures in an unsupervised way. Some modern approaches to novel data generation such as generative adversarial networks askew this symmetry, but still employ a pair of massive networks--one to generate the image and another to judge the images quality based on priors learned from a training set. This work introduces a chart autoencoder with an asymmetric encoding-decoding process that can incorporate additional semi-supervised information such as class labels. Besides enhancing the capability for handling data with complicated topological and geometric structures, the proposed model can successfully differentiate nearby but disjoint manifolds and intersecting manifolds with only a small amount of supervision. Moreover, this model only requires a low-complexity encoding operation, such as a locally defined linear projection. We discuss the approximation power of such networks and derive a bound that essentially depends on the intrinsic dimension of the data manifold rather than the dimension of ambient space. Next we incorporate bounds for the sampling rate of training data need to faithfully represent a given data manifold. We present numerical experiments that verify that the proposed model can effectively manage data with multi-class nearby but disjoint manifolds of different classes, overlapping manifolds, and manifolds with non-trivial topology. Finally, we conclude with some experiments on computer vision and molecular dynamics problems which showcase the efficacy of our methods on real-world data.

翻译：自编码是表示学习中的一种常用方法。传统的自编码器采用对称的编码-解码流程和简单的欧几里得潜在空间，以无监督方式检测隐藏的低维结构。一些现代新数据生成方法（如生成对抗网络）打破了这种对称性，但仍使用一对庞大的网络——一个用于生成图像，另一个根据从训练集学到的先验知识判断图像质量。本研究提出了一种具有非对称编码-解码过程的图册自编码器，能够融合类别标签等额外的半监督信息。该模型不仅能增强处理具有复杂拓扑和几何结构数据的能力，还能仅通过少量监督成功区分邻近但不相交的流形以及相交流形。此外，该模型仅需低复杂度的编码操作，例如局部定义的线性投影。我们讨论了此类网络的近似能力，并推导出一个本质上取决于数据流形本征维度而非环境空间维度的边界条件。接着我们整合了训练数据采样率所需的边界条件，以忠实表示给定数据流形。数值实验表明，所提模型能有效处理多类邻近但不相交的异类流形、重叠流形以及具有非平凡拓扑的流形。最后，我们在计算机视觉和分子动力学问题上进行了实验，展示了该方法在真实数据上的有效性。