## Abstract

We consider the problem of recovering random graph signals from nonlinear measurements. For this setting, closed-form Bayesian estimators are usually intractable and even numerical evaluation may be difficult to compute for large networks. In this paper, we propose a graph signal processing (GSP) framework for random graph signal recovery that utilizes information on 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 an approximate parametrization of the GSP-LMMSE estimator by shift-invariant graph filters by solving a weighted least-squares (WLS) problem. We present three implementations of the parametric GSP-LMMSE estimator for typical graph filters. These 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.

Original language | English |
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Pages (from-to) | 2207-2223 |

Number of pages | 17 |

Journal | IEEE Transactions on Signal Processing |

Volume | 70 |

DOIs | |

State | Accepted/In press - 1 Jan 2022 |

## Keywords

- Bayesian estimation
- Covariance matrices
- Estimation
- Filtering theory
- graph filters
- Graph signal processing (GSP)
- graph signal recovery
- GSP-LMMSE estimator
- Information filters
- linear minimum mean-squared-error (LMMSE) estimator
- Maximum likelihood detection
- Nonlinear filters
- sample-LMMSE estimator
- Training

## ASJC Scopus subject areas

- Signal Processing
- Electrical and Electronic Engineering