Learning Graphical Factor Models with Riemannian Optimization

TL;DR

This paper introduces a novel Riemannian optimization framework for learning graphical factor models with low-rank covariance structures, effectively handling heavy-tailed data in multivariate analysis.

Contribution

It proposes a flexible algorithmic approach combining graphical models and factor analysis under low-rank constraints using Riemannian optimization techniques.

Findings

Effective in modeling heavy-tailed distributions

Improves graph learning with low-rank covariance constraints

Demonstrates superior performance on real-world data

Abstract

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.