  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. 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, these models 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 encoder, such as local linear projection. We discuss
the theoretical approximation power of such networks that essentially depends
on the intrinsic dimension of the data manifold and not the dimension of the
observations. Our numerical experiments on synthetic and real-world data 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.
