  Constant-curvature Riemannian manifolds (CCMs) have been shown to be ideal
embedding spaces in many application domains, as their non-Euclidean geometry
can naturally account for some relevant properties of data, like hierarchy and
circularity. In this work, we introduce the CCM adversarial autoencoder
(CCM-AAE), a probabilistic generative model trained to represent a data
distribution on a CCM. Our method works by matching the aggregated posterior of
the CCM-AAE with a probability distribution defined on a CCM, so that the
encoder implicitly learns to represent data on the CCM to fool the
discriminator network. The geometric constraint is also explicitly imposed by
jointly training the CCM-AAE to maximise the membership degree of the
embeddings to the CCM. While a few works in recent literature make use of
either hyperspherical or hyperbolic manifolds for different learning tasks,
ours is the first unified framework to seamlessly deal with CCMs of different
curvatures. We show the effectiveness of our model on three different datasets
characterised by non-trivial geometry: semi-supervised classification on MNIST,
link prediction on two popular citation datasets, and graph-based molecule
generation using the QM9 chemical database. Results show that our method
improves upon other autoencoders based on Euclidean and non-Euclidean
geometries on all tasks taken into account.
