# Learning Graphs from Smooth and Graph-Stationary Signals with Hidden Variables

arXiv: https://arxiv.org/abs/2111.05588 · HTML: https://mlsanity.com/paper/2111.05588/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Nov 2021.

## Built on by (most cited first, 20 of 20)

- [Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals](https://mlsanity.com/paper/2404.02621/) (2024)
- [Non-negative Weighted DAG Structure Learning](https://mlsanity.com/paper/2409.07880/) (2024)
- [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)
- [Learning graphs and simplicial complexes from data](https://mlsanity.com/paper/2312.10545/) (2023)
- [Learning Cartesian Product Graphs with Laplacian Constraints](https://mlsanity.com/paper/2402.08105/) (2024)
- [Online Learning Of Expanding Graphs](https://mlsanity.com/paper/2409.08660/) (2024)
- [Network Topology Inference from Smooth Signals Under Partial Observability](https://mlsanity.com/paper/2410.05707/) (2024)
- [GKNet: Graph Kalman Filtering and Model Inference via Model-based Deep Learning](https://mlsanity.com/paper/2506.22004/) (2025)
- [Graph Convolutional Neural Networks Sensitivity under Probabilistic Error Model](https://mlsanity.com/paper/2203.07831/) (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)
- [Enhancing Graphical Lasso: A Robust Scheme for Non-Stationary Mean Data](https://mlsanity.com/paper/2503.19651/) (2025)
- [Learning Graph from Smooth Signals under Partial Observation: A Robustness Analysis](https://mlsanity.com/paper/2509.14887/) (2025)
- [Learning Kronecker-Structured Graphs from Smooth Signals](https://mlsanity.com/paper/2505.09822/) (2025)
- [Estimating Fair Graphs from Graph-Stationary Data](https://mlsanity.com/paper/2510.07536/) (2025)
- [Learning Time-Varying Graphs from Incomplete Graph Signals](https://mlsanity.com/paper/2510.17903/) (2025)
- [Non-negative DAG Learning from Time-Series Data](https://mlsanity.com/paper/2512.07267/) (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)

Network-topology inference from (vertex) signal observations is a prominent problem across data-science and engineering disciplines. Most existing schemes assume that observations from all nodes are available, but in many practical environments, only a subset of nodes is accessible. A natural (and sometimes effective) approach is to disregard the role of unobserved nodes, but this ignores latent network effects, deteriorating the quality of the estimated graph. Differently, this paper investigates the problem of inferring the topology of a network from nodal observations while taking into account the presence of hidden (latent) variables. Our schemes assume the number of observed nodes is considerably larger than the number of hidden variables and build on recent graph signal processing models to relate the signals and the underlying graph. Specifically, we go beyond classical correlation and partial correlation approaches and assume that the signals are smooth and/or stationary in the sought graph. The assumptions are codified into different constrained optimization problems, with the presence of hidden variables being explicitly taken into account. Since the resulting problems are ill-conditioned and non-convex, the block matrix structure of the proposed formulations is leveraged and suitable convex-regularized relaxations are presented. Numerical experiments over synthetic and real-world datasets showcase the performance of the developed methods and compare them with existing alternatives.