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 two new estimators that are both based on the Gauss-Newton method: 1) the elementwise graph-frequency-domain MAP (eGFD-MAP) estimator; and 2) the graph signal processing MAP (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 an ad-hoc method that minimizes the MAP objective function in the graph frequency domain and neglects mixed-derivatives of different graph frequencies in the Jacobian matrix as well as off-diagonal elements in the covariance matrices. Consequently, it updates the elements of the graph signal independently, which reduces the computational complexity compared to the conventional MAP estimator. The GSP-MAP estimator is based on optimizing the graph filters at each iteration of the Gauss-Newton algorithm. We state conditions under which the eGFD-MAP and GSP- MAP 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 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 iterative algorithms.

翻译：本文研究从非线性测量中恢复随机图信号的问题。我们构建了最大后验概率估计器，该估计器导致非凸优化问题。传统的非凸问题迭代求解方法对初始值敏感，计算复杂度高，且未利用数据背后的图结构。本文提出两种基于高斯-牛顿法的新估计器：1）逐元素图频域最大后验估计器；2）图信号处理最大后验估计器。每次迭代中，这些估计器通过两个图滤波器的输出进行更新，其中前一时刻的状态估计和残差作为输入图信号。逐元素图频域最大后验估计器是一种启发式方法，在图像频域中最小化最大后验目标函数，忽略雅可比矩阵中不同图频率的混合导数以及协方差矩阵中的非对角元素。因此，它独立更新图信号的每个元素，相比传统最大后验估计器降低了计算复杂度。图信号处理最大后验估计器基于在高斯-牛顿算法每次迭代中优化图滤波器。我们给出了在观测模型具有正交图频率的情况下，逐元素图频域最大后验估计器和图信号处理最大后验估计器与最大后验估计器一致的条件。通过合成数据和电力系统状态估计的真实问题，评估了这些估计器在非线性图信号恢复任务中的性能。仿真结果表明，所提估计器在计算复杂度、均方误差和对迭代算法初始值的鲁棒性方面具有优势。