# Stationary signal processing on graphs

arXiv: https://arxiv.org/abs/1601.02522 · HTML: https://mlsanity.com/paper/1601.02522/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jan 2016.

## Built on by (most cited first, 29 of 29)

- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Wiener Graph Deconvolutional Network Improves Graph Self-Supervised Learning](https://mlsanity.com/paper/2206.12933/) (2022)
- [Synthesizing Decentralized Controllers with Graph Neural Networks and Imitation Learning](https://mlsanity.com/paper/2012.14906/) (2020)
- [Gegenbauer Graph Neural Networks for Time-varying Signal Reconstruction](https://mlsanity.com/paper/2403.19800/) (2024)
- [Spatiotemporal Covariance Neural Networks](https://mlsanity.com/paper/2409.10068/) (2024)
- [Polynomial Graphical Lasso: Learning Edges from Gaussian Graph-Stationary Signals](https://mlsanity.com/paper/2404.02621/) (2024)
- [Node-Variant Graph Filters in Graph Neural Networks](https://mlsanity.com/paper/2106.00089/) (2021)
- [Network Topology Inference from Smooth Signals Under Partial Observability](https://mlsanity.com/paper/2410.05707/) (2024)
- [Missing Data in Signal Processing and Machine Learning: Models, Methods and Modern Approaches](https://mlsanity.com/paper/2506.01696/) (2025)
- [Generative Diffusion Models of Stochastic Graph Signals](https://mlsanity.com/paper/2607.06833/) (2026)
- [Propagation on Multi-relational Graphs for Node Regression](https://mlsanity.com/paper/2110.08185/) (2021)
- [Learning Graph ARMA Processes from Time-Vertex Spectra](https://mlsanity.com/paper/2302.06887/) (2023)
- [LogSpecT: Feasible Graph Learning Model from Stationary Signals with Recovery Guarantees](https://mlsanity.com/paper/2305.01379/) (2023)
- [Barron Space for Graph Convolution Neural Networks](https://mlsanity.com/paper/2311.02838/) (2023)
- [Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes](https://mlsanity.com/paper/2602.03055/) (2026)
- [Statistical Testing on Directed Graphs by Surrogate Data Generation](https://mlsanity.com/paper/2606.00758/) (2026)
- [Kernel-Based Models for Influence Maximization on Graphs based on Gaussian Process Variance Minimization](https://mlsanity.com/paper/2103.01575/) (2021)
- [Online non-parametric change-point detection for heterogeneous data streams observed over graph nodes](https://mlsanity.com/paper/2110.10518/) (2021)
- [Graph Filter Transfer via Probability Density Ratio Weighting](https://mlsanity.com/paper/2210.14633/) (2022)
- [Locally Stationary Graph Processes](https://mlsanity.com/paper/2309.01657/) (2023)
- [Sampling Transferable Graph Neural Networks with Limited Graph Information](https://mlsanity.com/paper/2410.16593/) (2024)
- [CoVariance Filters and Neural Networks over Hilbert Spaces](https://mlsanity.com/paper/2509.13178/) (2025)
- [Sampling Method for Generalized Graph Signals with Pre-selected Vertices via DC Optimization](https://mlsanity.com/paper/2509.14836/) (2025)
- [A Generative Model for Controllable Feature Heterophily in Graphs](https://mlsanity.com/paper/2509.23230/) (2025)
- [Estimating Fair Graphs from Graph-Stationary Data](https://mlsanity.com/paper/2510.07536/) (2025)
- [Decorrelating the Future: Joint Frequency Domain Learning for Spatio-temporal Forecasting](https://mlsanity.com/paper/2603.04418/) (2026)
- [Filter Learning for Subgraphs: Algebras and Performance Risk Bounds](https://mlsanity.com/paper/2607.21263/) (2026)
- [Role-Specific Predictive Geometries for Nonstationary Multivariate Graph-Signal Forecasting](https://mlsanity.com/paper/2609.06519/) (2026)
- [Colored Graph Signal Diffusion Models](https://mlsanity.com/paper/2610.17282/) (2026)

## Abstract (arXiv)

Graphs are a central tool in machine learning and information processing as they allow to conveniently capture the structure of complex datasets. In this context, it is of high importance to develop flexible models of signals defined over graphs or networks. In this paper, we generalize the traditional concept of wide sense stationarity to signals defined over the vertices of arbitrary weighted undirected graphs. We show that stationarity is expressed through the graph localization operator reminiscent of translation. We prove that stationary graph signals are characterized by a well-defined Power Spectral Density that can be efficiently estimated even for large graphs. We leverage this new concept to derive Wiener-type estimation procedures of noisy and partially observed signals and illustrate the performance of this new model for denoising and regression.