MAP Estimation of Graph Signals

In this paper, we consider the problem of recovering random graph signals from nonlinear measurements. We formulate the maximum a-posteriori probability (MAP) estimator, which results in a nonconvex optimization problem. Conventional iterative methods for minimizing nonconvex problems are sensitive to the initialization, have high computational complexity, and do not utilize the underlying graph structure behind the data. In this paper we propose three new estimators that are based on the Gauss-Newton method: 1) the elementwise graph-frequency-domain MAP (eGFD-MAP) estimator; 2) the sample graph signal processing MAP (sGSP-MAP) estimator; and 3) the GSP-MAP estimator. At each iteration, these estimators are updated by the outputs of two graph filters, with the previous state estimator and the residual as the input graph signals. The eGFD-MAP estimator is based on neglecting the mixed-derivatives of different graph frequencies in the Jacobian matrix and the off-diagonal elements in the covariance matrices. Consequently, it updates the elements of the graph signal in the graph-frequency domain independently, which reduces the computational complexity compared to the conventional MAP estimator. The sGSP-MAP and GSP-MAP estimators are based on optimizing the graph filters at each iteration of the Gauss-Newton algorithm. We state conditions under which the new estimators coincide with the MAP estimator in the case of an observation model with orthogonal graph frequencies. We evaluate the performance of the estimators for nonlinear graph signal recovery tasks, both with synthetic data and with the real-world problem of state estimation in power systems. These simulations show the advantages of the proposed estimators in terms of computational complexity, mean-squared-error, and robustness to the initialization of the algorithms.