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.

中文翻译：

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图形信号的基于 GSP 的 MAP 估计

在本文中，我们考虑从非线性测量中恢复随机图信号的问题。我们制定了最大后验概率 (MAP) 估计器，这会导致非凸优化问题。用于最小化非凸问题的传统迭代方法对初始化敏感，计算复杂度高，并且不利用数据背后的底层图结构。在本文中，我们提出了两个基于 Gauss-Newton 方法的新估计器：1）元素级图频域 MAP（eGFD-MAP）估计器；2）图形信号处理MAP（GSP-MAP）估计器。在每次迭代中，这些估计器由两个图滤波器的输出更新，前一个状态估计器和残差作为输入图信号。eGFD-MAP 估计器是一种特殊方法，它最小化图频域中的 MAP 目标函数，并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线元素。因此，它独立地更新了图信号的元素，与传统的 MAP 估计器相比，这降低了计算复杂度。GSP-MAP 估计器基于在 Gauss-Newton 算法的每次迭代中优化图滤波器。在具有正交图频率的观察模型的情况下，我们陈述了 eGFD-MAP 和 GSP-MAP 估计量与 MAP 估计量一致的条件。我们使用合成数据和电力系统中状态估计的现实问题评估非线性图信号恢复任务的估计器的性能。这些模拟显示了所提出的估计器在计算复杂性、均方误差和迭代算法初始化的鲁棒性方面的优势。