  Manifold-valued data naturally arises in medical imaging. In cognitive
neuroscience, for instance, brain connectomes base the analysis of coactivation
patterns between different brain regions on the analysis of the correlations of
their functional Magnetic Resonance Imaging (fMRI) time series - an object thus
constrained by construction to belong to the manifold of symmetric positive
definite matrices. One of the challenges that naturally arises consists of
finding a lower-dimensional subspace for representing such manifold-valued
data. Traditional techniques, like principal component analysis, are
ill-adapted to tackle non-Euclidean spaces and may fail to achieve a
lower-dimensional representation of the data - thus potentially pointing to the
absence of lower-dimensional representation of the data. However, these
techniques are restricted in that: (i) they do not leverage the assumption that
the connectomes belong on a pre-specified manifold, therefore discarding
information; (ii) they can only fit a linear subspace to the data. In this
paper, we are interested in variants to learn potentially highly curved
submanifolds of manifold-valued data. Motivated by the brain connectomes
example, we investigate a latent variable generative model, which has the added
benefit of providing us with uncertainty estimates - a crucial quantity in the
medical applications we are considering. While latent variable models have been
proposed to learn linear and nonlinear spaces for Euclidean data, or geodesic
subspaces for manifold data, no intrinsic latent variable model exists to learn
nongeodesic subspaces for manifold data. This paper fills this gap and
formulates a Riemannian variational autoencoder with an intrinsic generative
model of manifold-valued data. We evaluate its performances on synthetic and
real datasets by introducing the formalism of weighted Riemannian submanifolds.
