# Joint Inference of Multiple Graphs from Matrix Polynomials

arXiv: https://arxiv.org/abs/2010.08120 · HTML: https://mlsanity.com/paper/2010.08120/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2020.

## Built on by (most cited first, 19 of 19)

- [Joint Network Topology Inference via a Shared Graphon Model](https://mlsanity.com/paper/2209.08223/) (2022)
- [Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals](https://mlsanity.com/paper/2404.02621/) (2024)
- [Joint inference of multiple graphs with hidden variables from stationary graph signals](https://mlsanity.com/paper/2110.03666/) (2021)
- [Learning Product Graphs from Spectral Templates](https://mlsanity.com/paper/2211.02893/) (2022)
- [Joint graph learning from Gaussian observations in the presence of hidden nodes](https://mlsanity.com/paper/2212.01816/) (2022)
- [Graph Laplacian Learning with Exponential Family Noise](https://mlsanity.com/paper/2306.08201/) (2023)
- [Multiview Graph Learning with Consensus Graph](https://mlsanity.com/paper/2401.13769/) (2024)
- [Learning Cartesian Product Graphs with Laplacian Constraints](https://mlsanity.com/paper/2402.08105/) (2024)
- [Joint Network Topology Inference via Structured Fusion Regularization](https://mlsanity.com/paper/2103.03471/) (2021)
- [Graphon-aided Joint Estimation of Multiple Graphs](https://mlsanity.com/paper/2202.05686/) (2022)
- [Online Inference for Mixture Model of Streaming Graph Signals with Non-White Excitation](https://mlsanity.com/paper/2207.14019/) (2022)
- [LogSpecT: Feasible Graph Learning Model from Stationary Signals with Recovery Guarantees](https://mlsanity.com/paper/2305.01379/) (2023)
- [Online Network Inference from Graph-Stationary Signals with Hidden Nodes](https://mlsanity.com/paper/2409.08760/) (2024)
- [Graph-Dictionary Signal Model for Sparse Representations of Multivariate Data](https://mlsanity.com/paper/2411.05729/) (2024)
- [On the Stability of Graph Convolutional Neural Networks: A Probabilistic Perspective](https://mlsanity.com/paper/2506.01213/) (2025)
- [Estimating Fair Graphs from Graph-Stationary Data](https://mlsanity.com/paper/2510.07536/) (2025)
- [Co-Hub Node Based Multiview Graph Learning with Theoretical Guarantees](https://mlsanity.com/paper/2512.12435/) (2025)
- [Directed Graph Topology Inference via Graph Filter Identification](https://mlsanity.com/paper/2606.27455/) (2026)
- [From topology learning to graph generation: A unifying perspective](https://mlsanity.com/paper/2609.02286/) (2026)

## Abstract (arXiv)

Inferring graph structure from observations on the nodes is an important and popular network science task. Departing from the more common inference of a single graph and motivated by social and biological networks, we study the problem of jointly inferring multiple graphs from the observation of signals at their nodes (graph signals), which are assumed to be stationary in the sought graphs. From a mathematical point of view, graph stationarity implies that the mapping between the covariance of the signals and the sparse matrix representing the underlying graph is given by a matrix polynomial. A prominent example is that of Markov random fields, where the inverse of the covariance yields the sparse matrix of interest. From a modeling perspective, stationary graph signals can be used to model linear network processes evolving on a set of (not necessarily known) networks. Leveraging that matrix polynomials commute, a convex optimization method along with sufficient conditions that guarantee the recovery of the true graphs are provided when perfect covariance information is available. Particularly important from an empirical viewpoint, we provide high-probability bounds on the recovery error as a function of the number of signals observed and other key problem parameters. Numerical experiments using synthetic and real-world data demonstrate the effectiveness of the proposed method with perfect covariance information as well as its robustness in the noisy regime.