  With the recent advance of geometric deep learning, neural networks have been
extensively used for data in non-Euclidean domains. In particular, hyperbolic
neural networks have proved successful in processing hierarchical information
of data. However, many hyperbolic neural networks are numerically unstable
during training, which precludes using complex architectures. This crucial
problem makes it difficult to build hyperbolic generative models for real and
complex data. In this work, we propose a hyperbolic generative network in which
we design novel architecture and layers to improve stability in training. Our
proposed network contains three parts: first, a hyperbolic autoencoder (AE)
that produces hyperbolic embedding for input data; second, a hyperbolic
generative adversarial network (GAN) for generating the hyperbolic latent
embedding of the AE from simple noise; third, a generator that inherits the
decoder from the AE and the generator from the GAN. We call this network the
hyperbolic AE-GAN, or HAEGAN for short. The architecture of HAEGAN fosters
expressive representation in the hyperbolic space, and the specific design of
layers ensures numerical stability. Experiments show that HAEGAN is able to
generate complex data with state-of-the-art structure-related performance.
