  Graphical models and factor analysis are well-established tools in
multivariate statistics. While these models can be both linked to structures
exhibited by covariance and precision matrices, they are generally not jointly
leveraged within graph learning processes. This paper therefore addresses this
issue by proposing a flexible algorithmic framework for graph learning under
low-rank structural constraints on the covariance matrix. The problem is
expressed as penalized maximum likelihood estimation of an elliptical
distribution (a generalization of Gaussian graphical models to possibly
heavy-tailed distributions), where the covariance matrix is optionally
constrained to be structured as low-rank plus diagonal (low-rank factor model).
The resolution of this class of problems is then tackled with Riemannian
optimization, where we leverage geometries of positive definite matrices and
positive semi-definite matrices of fixed rank that are well suited to
elliptical models. Numerical experiments on real-world data sets illustrate the
effectiveness of the proposed approach.
