In this paper, we consider the problem of recovering random graph signals with complex values. For general Bayesian estimation of complex-valued vectors, it is known that the widely-linear minimum mean-squared-error (WLMMSE) estimator can achieve a lower mean-squared-error (MSE) than that of the linear minimum MSE (LMMSE) estimator. Inspired by the WLMMSE estimator, in this paper we develop the graph signal processing (GSP)-WLMMSE estimator, which minimizes the MSE among estimators that are represented as a two-channel output of a graph filter, i.e. widely-linear GSP estimators. We discuss the properties of the proposed GSP-WLMMSE estimator. In particular, we show that the MSE of the GSP-WLMMSE estimator is always equal to or lower than the MSE of the GSP-LMMSE estimator. The GSP-WLMMSE estimator is based on diagonal covariance matrices in the graph frequency domain, and thus has reduced complexity compared with the WLMMSE estimator. This property is especially important when using the sample-mean versions of these estimators that are based on a training dataset. We then state conditions under which the low-complexity GSP-WLMMSE estimator coincides with the WLMMSE estimator. In the simulations, we investigate two synthetic estimation problems (with linear and nonlinear models) and the problem of state estimation in power systems. For these problems, it is shown that the GSP-WLMMSE estimator outperforms the GSP-LMMSE estimator and achieves similar performance to that of the WLMMSE estimator.

翻译：本文研究复数值随机图信号的恢复问题。对于复数值向量的广义贝叶斯估计，已知广义线性最小均方误差（WLMMSE）估计器能够达到比线性最小均方误差（LMMSE）估计器更低的均方误差（MSE）。受WLMMSE估计器的启发，本文提出了图信号处理（GSP）-WLMMSE估计器，该估计器在可表示为图滤波器双通道输出的估计器（即广义线性GSP估计器）中最小化MSE。我们讨论了所提出的GSP-WLMMSE估计器的性质。特别地，我们证明了GSP-WLMMSE估计器的MSE始终小于或等于GSP-LMMSE估计器的MSE。GSP-WLMMSE估计器基于图频域中的对角协方差矩阵，因此与WLMMSE估计器相比具有更低的复杂度。当使用基于训练数据集的样本均值版本估计器时，这一特性尤为重要。然后我们给出了低复杂度GSP-WLMMSE估计器与WLMMSE估计器一致的条件。在仿真中，我们研究了两个合成估计问题（具有线性和非线性模型）以及电力系统中的状态估计问题。结果表明，对于这些问题，GSP-WLMMSE估计器优于GSP-LMMSE估计器，并实现了与WLMMSE估计器相似的性能。