  Variational auto-encoders (VAEs) have proven to be a well suited tool for
performing dimensionality reduction by extracting latent variables lying in a
potentially much smaller dimensional space than the data. Their ability to
capture meaningful information from the data can be easily apprehended when
considering their capability to generate new realistic samples or perform
potentially meaningful interpolations in a much smaller space. However, such
generative models may perform poorly when trained on small data sets which are
abundant in many real-life fields such as medicine. This may, among others,
come from the lack of structure of the latent space, the geometry of which is
often under-considered. We thus propose in this paper to see the latent space
as a Riemannian manifold endowed with a parametrized metric learned at the same
time as the encoder and decoder networks. This metric is then used in what we
called the Riemannian Hamiltonian VAE which extends the Hamiltonian VAE
introduced by arXiv:1805.11328 to better exploit the underlying geometry of the
latent space. We argue that such latent space modelling provides useful
information about its underlying structure leading to far more meaningful
interpolations, more realistic data-generation and more reliable clustering.
