# Bayesian Estimation of Graph Signals

arXiv: https://arxiv.org/abs/2103.15520 · HTML: https://mlsanity.com/paper/2103.15520/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Mar 2021.

## Built on by (most cited first, 9 of 9)

- [GSP-KalmanNet: Tracking Graph Signals via Neural-Aided Kalman Filtering](https://mlsanity.com/paper/2311.16602/) (2023)
- [Graph-Aware Diffusion for Signal Generation](https://mlsanity.com/paper/2510.05036/) (2025)
- [Efficient Graph Laplacian Estimation by Proximal Newton](https://mlsanity.com/paper/2302.06434/) (2023)
- [Learning Graph Filters for Structure-Function Coupling based Hub Node Identification](https://mlsanity.com/paper/2410.17410/) (2024)
- [Graph Distribution-valued Signals: A Wasserstein Space Perspective](https://mlsanity.com/paper/2509.25802/) (2025)
- [Graph Filter Transfer via Probability Density Ratio Weighting](https://mlsanity.com/paper/2210.14633/) (2022)
- [Cramer-Rao Bounds for Laplacian Matrix Estimation](https://mlsanity.com/paper/2504.04576/) (2025)
- [Rotation-Parameterized Graph Fractional Fourier Transform: Definition, Properties, and Optimal Filtering](https://mlsanity.com/paper/2511.16111/) (2025)
- [Filter Learning for Subgraphs: Algebras and Performance Risk Bounds](https://mlsanity.com/paper/2607.21263/) (2026)

## Abstract (arXiv)

We consider the problem of recovering random graph signals from nonlinear measurements. For this case, closed-form Bayesian estimators are usually intractable and even numerical evaluation of these estimators may be hard to compute for large networks. In this paper, we propose a graph signal processing (GSP) framework for random graph signal recovery that utilizes the information of the structure behind the data. First, we develop the GSP-linear minimum mean-squared-error (GSP-LMMSE) estimator, which minimizes the mean-squared error (MSE) among estimators that are represented as an output of a graph filter. The GSP-LMMSE estimator is based on diagonal covariance matrices in the graph frequency domain, and thus, has reduced complexity compared with the LMMSE estimator. This property is especially important when using the sample-mean versions of these estimators that are based on a training dataset. We then state conditions under which the low-complexity GSP-LMMSE estimator coincides with the optimal LMMSE estimator. Next, we develop the approximated parametrization of the GSP-LMMSE estimator by shift-invariant graph filters by solving a weighted least squared (WLS) problem. We present three implementations of the parametric GSP-LMMSE estimator for typical graph filters. Parametric graph filters are more robust to outliers and to network topology changes. In our simulations, we evaluate the performance of the proposed GSP-LMMSE estimators for the problem of state estimation in power systems, which can be interpreted as a graph signal recovery task. We show that the proposed sample-GSP estimators outperform the sample-LMMSE estimator for a limited training dataset and that the parametric GSP-LMMSE estimators are more robust to topology changes in the form of adding/removing vertices/edges.