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.

翻译：图形模型与因子分析是多元统计学中成熟的分析工具。尽管这两类模型均可关联协方差矩阵与精度矩阵所呈现的结构特征，但在图学习过程中通常未被联合利用。为此，本文提出一种灵活的算法框架，在协方差矩阵低秩结构约束下实现图学习。该问题被表述为椭圆分布（高斯图形模型向可能具有重尾分布情形的泛化）的带罚极大似然估计，其中协方差矩阵可被约束为"低秩矩阵加对角矩阵"结构（低秩因子模型）。针对此类问题，我们采用黎曼优化方法求解，利用正定矩阵与固定秩半正定矩阵的黎曼几何结构（该结构特别适配椭圆模型）。在真实数据集上的数值实验验证了所提方法的有效性。