Bayesian Estimation of Graph Signals
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) fr...
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| Published in | IEEE transactions on signal processing Vol. 70; pp. 2207 - 2223 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
IEEE
2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1053-587X 1941-0476 |
| DOI | 10.1109/TSP.2022.3159393 |
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| 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 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 graph filters. 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. |
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| AbstractList | 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 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 graph filters. 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. |
| Author | Eldar, Yonina C. Routtenberg, Tirza Kroizer, Ariel |
| Author_xml | – sequence: 1 givenname: Ariel surname: Kroizer fullname: Kroizer, Ariel email: arielkro@post.bgu.ac.il organization: School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel – sequence: 2 givenname: Tirza orcidid: 0000-0002-7238-7764 surname: Routtenberg fullname: Routtenberg, Tirza email: tirzar@bgu.ac.il organization: School of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva, Israel – sequence: 3 givenname: Yonina C. surname: Eldar fullname: Eldar, Yonina C. email: yonina@weizmann.ac.il organization: Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel |
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| SubjectTerms | Apexes Bayesian analysis Bayesian estimation Complexity Covariance matrices Covariance matrix Datasets Estimation Estimators Filtering theory graph filters Graph signal processing (GSP) graph signal recovery Graph theory Graphical representations GSP-LMMSE estimator Information filters linear minimum mean-squared-error (LMMSE) estimator Maximum likelihood detection Network topologies Nonlinear filters Outliers (statistics) Parameterization Performance evaluation Robustness (mathematics) sample-LMMSE estimator Signal processing Signal reconstruction State estimation Training |
| Title | Bayesian Estimation of Graph Signals |
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