# Joint graph learning from Gaussian observations in the presence of hidden nodes

arXiv: https://arxiv.org/abs/2212.01816 · HTML: https://mlsanity.com/paper/2212.01816/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 39 references that are in the map.
- First posted on arXiv 4 Dec 2022.

## Builds on (most cited first, 17 of 17)

- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [Graph Learning: A Survey](https://mlsanity.com/paper/2105.00696/) (2021)
- [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)
- [Inference for multiple heterogeneous networks with a common invariant subspace](https://mlsanity.com/paper/1906.10026/) (2019)
- [Structured Graph Learning Via Laplacian Spectral Constraints](https://mlsanity.com/paper/1909.11594/) (2019)
- [Learning Graphs from Smooth and Graph-Stationary Signals with Hidden Variables](https://mlsanity.com/paper/2111.05588/) (2021)
- [Online Topology Inference from Streaming Stationary Graph Signals with Partial Connectivity Information](https://mlsanity.com/paper/2007.03653/) (2020)
- [Joint Inference of Multiple Graphs from Matrix Polynomials](https://mlsanity.com/paper/2010.08120/) (2020)
- [Accelerated Graph Learning from Smooth Signals](https://mlsanity.com/paper/2110.09677/) (2021)
- [Robust Graph Filter Identification and Graph Denoising from Signal Observations](https://mlsanity.com/paper/2210.08488/) (2022)
- [Enhanced graph-learning schemes driven by similar distributions of motifs](https://mlsanity.com/paper/2207.04747/) (2022)
- [Joint Network Topology Inference via a Shared Graphon Model](https://mlsanity.com/paper/2209.08223/) (2022)
- [Graph topology inference based on sparsifying transform learning](https://mlsanity.com/paper/1806.01701/) (2018)
- [Joint inference of multiple graphs with hidden variables from stationary graph signals](https://mlsanity.com/paper/2110.03666/) (2021)
- [Multilayer weighted social network model](https://mlsanity.com/paper/1408.1190/) (2014)

## Built on by (most cited first, 3 of 3)

- [Fair GLASSO: Estimating Fair Graphical Models with Unbiased Statistical Behavior](https://mlsanity.com/paper/2406.09513/) (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)

## Abstract (arXiv)

Graph learning problems are typically approached by focusing on learning the topology of a single graph when signals from all nodes are available. However, many contemporary setups involve multiple related networks and, moreover, it is often the case that only a subset of nodes is observed while the rest remain hidden. Motivated by this, we propose a joint graph learning method that takes into account the presence of hidden (latent) variables. Intuitively, the presence of the hidden nodes renders the inference task ill-posed and challenging to solve, so we overcome this detrimental influence by harnessing the similarity of the estimated graphs. To that end, we assume that the observed signals are drawn from a Gaussian Markov random field with latent variables and we carefully model the graph similarity among hidden (latent) nodes. Then, we exploit the structure resulting from the previous considerations to propose a convex optimization problem that solves the joint graph learning task by providing a regularized maximum likelihood estimator. Finally, we compare the proposed algorithm with different baselines and evaluate its performance over synthetic and real-world graphs.