  Deep probabilistic generative models have achieved incredible success in many
fields of application. Among such models, variational autoencoders (VAEs) have
proved their ability in modeling a generative process by learning a latent
representation of the input. In this paper, we propose a novel VAE defined in
the quaternion domain, which exploits the properties of quaternion algebra to
improve performance while significantly reducing the number of parameters
required by the network. The success of the proposed quaternion VAE with
respect to traditional VAEs relies on the ability to leverage the internal
relations between quaternion-valued input features and on the properties of
second-order statistics which allow to define the latent variables in the
augmented quaternion domain. In order to show the advantages due to such
properties, we define a plain convolutional VAE in the quaternion domain and we
evaluate its performance with respect to its real-valued counterpart on the
CelebA face dataset.
