Leveraging Low-Rank Factorizations of Conditional Correlation Matrices in Graph Learning

Abstract

This paper addresses the problem of learning an undirected graph from data gathered at each node. Within Gaussian graphical models (GGM), the topology of such graph can be linked to the support of the conditional correlation matrix of the data. The corresponding graph learning problem then scales as the square of number of variables (nodes), which is usually problematic for large dimension. To tackle this issue, we propose a graph learning framework that leverages a low-rank factorization of the conditional correlation matrix. In order to solve the resulting optimization problem, we derive tools required to apply Riemannian optimization techniques for this particular structure. The proposal is then particularized to a low-rank constrained counterpart of the standard GGM estimation problem, i.e., the regularized maximum likelihood estimation of a precision matrix. Experiments on synthetic and real data demonstrate that a very efficient dimension-versus-performance trade-off can be achieved with this approach.