# Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals

arXiv: https://arxiv.org/abs/2404.02621 · HTML: https://mlsanity.com/paper/2404.02621/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 43 references that are in the map.
- First posted on arXiv 3 Apr 2024.

## Builds on (most cited first, 15 of 15)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Learning graphs from data: A signal representation perspective](https://mlsanity.com/paper/1806.00848/) (2018)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Network Topology Inference from Spectral Templates](https://mlsanity.com/paper/1608.03008/) (2016)
- [Stationary Graph Processes and Spectral Estimation](https://mlsanity.com/paper/1603.04667/) (2016)
- [Stationary signal processing on graphs](https://mlsanity.com/paper/1601.02522/) (2016)
- [Learning Graphs from Smooth and Graph-Stationary Signals with Hidden Variables](https://mlsanity.com/paper/2111.05588/) (2021)
- [Joint Inference of Multiple Graphs from Matrix Polynomials](https://mlsanity.com/paper/2010.08120/) (2020)
- [Elastic Net Regularization Paths for All Generalized Linear Models](https://mlsanity.com/paper/2103.03475/) (2021)
- [Algorithms for Learning Graphs in Financial Markets](https://mlsanity.com/paper/2012.15410/) (2020)
- [Learning Undirected Graphs in Financial Markets](https://mlsanity.com/paper/2005.09958/) (2020)
- [Graph topology inference based on sparsifying transform learning](https://mlsanity.com/paper/1806.01701/) (2018)
- [Graph Learning from Gaussian and Stationary Graph Signals](https://mlsanity.com/paper/2303.07041/) (2023)
- [Estimation of partially known Gaussian graphical models with score-based structural priors](https://mlsanity.com/paper/2401.14340/) (2024)
- [A Majorized ADMM with Indefinite Proximal Terms for Linearly Constrained Convex Composite Optimization](https://mlsanity.com/paper/1412.1911/) (2014)

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

- [Fair GLASSO: Estimating Fair Graphical Models with Unbiased Statistical Behavior](https://mlsanity.com/paper/2406.09513/) (2024)
- [Online Learning Of Expanding Graphs](https://mlsanity.com/paper/2409.08660/) (2024)
- [Online Network Inference from Graph-Stationary Signals with Hidden Nodes](https://mlsanity.com/paper/2409.08760/) (2024)
- [Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data](https://mlsanity.com/paper/2503.19651/) (2025)
- [Leveraging Low-rank Factorizations of Conditional Correlation Matrices in Graph Learning](https://mlsanity.com/paper/2506.10628/) (2025)
- [Estimating Fair Graphs from Graph-Stationary Data](https://mlsanity.com/paper/2510.07536/) (2025)
- [Comprehensive Robust Dynamic Mode Decomposition from Mode Extraction to Dimensional Reduction](https://mlsanity.com/paper/2601.11116/) (2026)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)

## Abstract (arXiv)

This paper introduces Polynomial Graphical Lasso (PGL), a new approach to learning graph structures from nodal signals. Our key contribution lies in modeling the signals as Gaussian and stationary on the graph, enabling the development of a graph-learning formulation that combines the strengths of graphical lasso with a more encompassing model. Specifically, we assume that the precision matrix can take any polynomial form of the sought graph, allowing for increased flexibility in modeling nodal relationships. Given the resulting complexity and nonconvexity of the resulting optimization problem, we (i) propose a low-complexity algorithm that alternates between estimating the graph and precision matrices, and (ii) characterize its convergence. We evaluate the performance of PGL through comprehensive numerical simulations using both synthetic and real data, demonstrating its superiority over several alternatives. Overall, this approach presents a significant advancement in graph learning and holds promise for various applications in graph-aware signal analysis and beyond.