Learning Graphical Factor Models with Riemannian Optimization

Graphical models and factor analysis are well-established tools in multivariate statistics. While these models can be both linked tostructures exhibited by covariance and precision matrices, they are generally not jointly leveraged within graph learning processes. This papertherefore addresses this issue by proposing a flexible algorithmic framework for graph learning under low-rank structural constraints on thecovariance 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-rankplus diagonal (low-rank factor model). The resolution of this class of problems is then tackled with Riemannian optimization, where we lever-age geometries of positive definite matrices and positive semi-definite matrices of fixed rank that are well suited to elliptical models. Numericalexperiments on synthetic and real-world data sets illustrate the effectiveness of the proposed approach