  Deep generative networks have been widely used for learning mappings from a
low-dimensional latent space to a high-dimensional data space. In many cases,
data transformations are defined by linear paths in this latent space. However,
the Euclidean structure of the latent space may be a poor match for the
underlying latent structure in the data. In this work, we incorporate a
generative manifold model into the latent space of an autoencoder in order to
learn the low-dimensional manifold structure from the data and adapt the latent
space to accommodate this structure. In particular, we focus on applications in
which the data has closed transformation paths which extend from a starting
point and return to nearly the same point. Through experiments on data with
natural closed transformation paths, we show that this model introduces the
ability to learn the latent dynamics of complex systems, generate
transformation paths, and classify samples that belong on the same
transformation path.
