# Learning Graphical Factor Models with Riemannian Optimization

arXiv: https://arxiv.org/abs/2210.11950 · HTML: https://mlsanity.com/paper/2210.11950/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 73 references that are in the map.
- First posted on arXiv 21 Oct 2022.

## Builds on (most cited first, 16 of 16)

- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Maximum likelihood estimation in Gaussian models under total positivity](https://mlsanity.com/paper/1702.04031/) (2017)
- [MADMM: a generic algorithm for non-smooth optimization on manifolds](https://mlsanity.com/paper/1505.07676/) (2015)
- [Graphical Lasso and Thresholding: Equivalence and Closed-form Solutions](https://mlsanity.com/paper/1708.09479/) (2017)
- [Estimation of Graphical Models through Structured Norm Minimization](https://mlsanity.com/paper/1609.09010/) (2016)
- [Learning Latent Variable Gaussian Graphical Models](https://mlsanity.com/paper/1406.2721/) (2014)
- [A Proximal Approach for a Class of Matrix Optimization Problems](https://mlsanity.com/paper/1801.07452/) (2018)
- [Fast Projected Newton-like Method for Precision Matrix Estimation under Total Positivity](https://mlsanity.com/paper/2112.01939/) (2021)
- [Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models](https://mlsanity.com/paper/1304.3206/) (2013)
- [Robust Graphical Modeling with t-Distributions](https://mlsanity.com/paper/1408.2033/) (2014)
- [On the inconsistency of ℓ1-penalised sparse precision matrix estimation](https://mlsanity.com/paper/1603.02532/) (2016)
- [Elliptical graphical modelling](https://mlsanity.com/paper/1506.04321/) (2015)
- [Computation of the Maximum Likelihood estimator in low-rank Factor Analysis](https://mlsanity.com/paper/1801.05935/) (2018)
- [Bayesian Scalable Precision Factor Analysis for Massive Sparse Gaussian Graphical Models](https://mlsanity.com/paper/2107.11316/) (2021)
- [New insights into the statistical properties of M-estimators](https://mlsanity.com/paper/1710.09945/) (2017)

## Built on by (most cited first, 8 of 8)

- [The Fisher-Rao geometry of CES distributions](https://mlsanity.com/paper/2310.01032/) (2023)
- [Sparse Graphical Linear Dynamical Systems](https://mlsanity.com/paper/2307.03210/) (2023)
- [Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure](https://mlsanity.com/paper/2406.09023/) (2024)
- [Elliptical Wishart distributions: information geometry, maximum likelihood estimator, performance analysis and statistical learning](https://mlsanity.com/paper/2411.02726/) (2024)
- [Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning](https://mlsanity.com/paper/2506.10628/) (2025)
- [T-Rex: Fitting a Robust Factor Model via Expectation-Maximization](https://mlsanity.com/paper/2505.12117/) (2025)
- [Including Node Textual Metadata in Laplacian-constrained Gaussian Graphical Models](https://mlsanity.com/paper/2602.15920/) (2026)
- [Dynamic Elliptical Graph Factor Models via Riemannian Optimization with Geodesic Temporal Regularization](https://mlsanity.com/paper/2605.18316/) (2026)

## Abstract (arXiv)

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.