论文标题
基于GSP的图形信号估计
GSP-Based MAP Estimation of Graph Signals
论文作者
论文摘要
在本文中，我们考虑了从非线性测量中恢复随机图信号的问题。我们制定了最大A-tosterii概率（MAP）估计量，这导致了非convex优化问题。最小化非凸问题的常规迭代方法对初始化敏感，具有较高的计算复杂性，并且不利用数据背后的基础图结构。在本文中，我们提出了两个基于高斯 - 纽顿方法的新估计量：1）元素定为图形频率域图（EGFD-MAP）估计器； 2）图形信号处理图（GSP-MAP）估计器。在每次迭代时，这些估计器将通过两个图形过滤器的输出进行更新，并以先前的状态估计器和残差为输入图信号。 EGFD-MAP估计器是一种临时方法，可将图频域中的MAP目标函数最小化，并忽略了Jacobian矩阵中不同图形频率的混合衍生物以及协和矩阵中的非对角线元素。因此，它独立地更新图形信号的元素，这与常规MAP估计器相比降低了计算复杂性。 GSP-MAP估计器基于在高斯 - 纽顿算法的每次迭代中优化图形过滤器。我们说明EGFD-MAP和GSP-图估计器与MAP估计器一致的条件，在观察模型的情况下，具有正交图频率。我们通过合成数据以及电力系统中的状态估计问题来评估非线性图信号恢复任务的估计器性能。这些模拟显示了所提出的估计量在计算复杂性，均值错误和对迭代算法初始化的鲁棒性方面的优势。
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.
